$f\circ{\cal S}_{2}=\exp\bigg{[}\left\\{\ldots,G\right\\}\bigg{]}f\equiv
f+\left\\{f,G\right\\}+\frac{1}{2}\left\\{\left\\{f,G\right\\},G\right\\}+\ldots$
(3.40)

Here the notation $\left\\{\ldots,G\right\\}$ means the differential operator
that acts on functionals $f$ and returns $\\{f,G\\}$. The definition (3.40)
uses the fact that a general symplectomorphism can be obtained by
exponentiating, as for any Lie group. The functional $G$ is similar to a
Hamilton-Jacobi functional in that it encodes the dynamics (except that it is
a functional of original coordinates and original momenta instead of being a
functional of original coordinates and new momenta).

The results of Secs. 5 and 6 below imply that $G=O(\alpha^{3})$, and it
follows that through $O(\alpha^{5})$ it is sufficient to use a truncated
version of the formula (3.40) that retains only the first two terms. In
particular, if $y^{a}$ are arbitrary coordinates on phase space, the
scattering map ${\cal S}_{2}$ can be represented by

$y^{a}\circ{\cal S}_{2}=y^{a}+\left\\{y^{a},G\right\\}+O(\alpha^{6}).$ (3.41)

In Appendix G we relate the generating functional $G$ to the various
functionals ${\cal H}^{\rm e}$, ${\cal H}^{\rm m}$ and ${\cal K}$ defined in
Sec. 3.5 above. We also show in Appendix G that $G$ has the specific form

$G\left[{\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}},{\hat{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}\right]={\mathscr{G}}\left[{\breve{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}},e^{-i{\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}}e^{i{\Delta\Psi}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}/2}\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}\right].$
(3.42)

Here on the left hand side the arguments

$\left({\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}},{\hat{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}\right)$ (3.43)

are the independent phase space coordinates defined in Sec. 2.5, on the right
hand side ${\mathscr{G}}$ is some functional, and its arguments are given in
terms of the fields on the left hand side by

$\displaystyle{\breve{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$})$
$\displaystyle=$ $\displaystyle{\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$})+g(u)\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$}),$ (3.44a)
$\displaystyle{\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$})$
$\displaystyle=$ $\displaystyle{\hat{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$})-2\int
dug^{\prime}(u){\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$}),$
(3.44b)

[cf. Eqs. (2.51) and (3.29)]. The functional ${\mathscr{G}}$ must obey the
constraint (G.10), whose interpretation is that its functional derivative with
respect to its first argument ${\breve{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ involves just a bulk term and no
boundary term. It must also obey the constraint (G.7). These constraints
together with the form (3.42) of the generating functional guarantee that the
corresponding symplectomorphism (3.40) satisfies the required conditions
discussed in Sec. 3.5: it preserves the conserved quantities (3.35) and (2.39)
and transforms appropriately under the symmetry (3.28).

##  4 Coupling of soft and hard degrees of freedom: general arguments and
their validity

We now turn to an assessment of some of the arguments that have been made in
the literature about the nature of the dynamics of the soft degrees of
freedom, using as a foundation properties of the scattering map derived in the
previous section. In the following three subsections, we will discuss three
different lines of reasoning for why the soft dynamics is either trivial or
completely decoupled from the dynamics of a hard sector. We will argue that
these conclusions are unfounded.

###  4.1 Argument for triviality of soft dynamics

One argument that might be made about the dynamics of the soft degrees of
freedom is that they are a trivial extension of the dynamics of the hard
degrees of freedom. Specifically, the full dynamics can be derived from the
generating functional [cf. Eq. (3.42) above]

${\mathscr{G}}\left[{\breve{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}},e^{-i{\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}}e^{i{\Delta\Psi}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}/2}\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}\right],$
(4.1)

where we recall that the definition (3.29) of the field ${\breve{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ requires the boundary condition

$\lim_{u\to\pm\infty}\,{\breve{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$})=\pm\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$})/2.$ (4.2)

Suppose now that one considers scattering with purely hard initial data, for
which ${\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}=\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}=0$141414This condition is
equivalent to ${\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}=\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}=0$, since we are working within a
framework where the $+$ and $-$ variables are related by the linear order
scattering map (3.23), cf. Eqs. (3.39) and (3.40) above.. Such scattering is
defined by a functional ${\mathscr{G}}[{\breve{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}]$,
and fully general scattering is defined by the same functional (4.1) except
that one of the arguments is modified by a phase factor. Therefore fully
general scattering is a trivial extension of purely hard scattering.

A potential loophole in this argument is the fact that the functionals in the
two different cases are defined on different function spaces. The purely hard
scattering functional is defined on a function space where
${\breve{\Psi}}^{\rm e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ satisfies the
boundary conditions (4.2) with vanishing right hand side, whereas the fully
general functional is defined on the larger function space where the right
hand side is nonzero. However, it turns out that within the context of
perturbative classical scattering, the general functional ${\mathscr{G}}$ is
determined by continuity by its restriction to the smaller function space.
This follows from the constraint (G.10) that the generating functional must
satisfy. See Sec. 5 below for further discussion of this point within the
context of second order perturbation theory. Thus the more general functional
is determined uniquely by the restricted functional.

There is, however, a flaw in the argument. Scattering with initial data with
${\bar{\Psi}}^{\rm e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}=\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}=0$ is not “purely hard” scattering,
since these variables will generically be nonzero after the scattering due to
the electromagnetic memory effect (see Sec. 6.2 and Appendix J below for more
details). Thus, even within the restricted functional, the hard and soft
degrees of freedom are intertwined.

###  4.2 Argument for decoupling based on theorem in symplectic geometry

A more fruitful approach, following Bousso and Porrati [33] (henceforth BP),
is to focus on the quantities that are preserved under the scattering map
(1.4), instead of on $\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ and ${\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$. These quantities are the charge
$Q_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$})$ given by
Eq. (3.24), and the quantity $\Psi^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{--}$}}(\mbox{\boldmath$\theta$})$
given by Eqs. (2.47) and (2.49) which is preserved by Eq. (2.39). From Eqs.
(2.52) these quantities have the Poisson brackets

$\displaystyle\left\\{Q_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$}),Q_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$}^{\prime})\right\\}$
$\displaystyle=$ $\displaystyle 0,$ (4.3a) $\displaystyle\left\\{\Psi^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{--}$}}(\mbox{\boldmath$\theta$}),\Psi^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{--}$}}(\mbox{\boldmath$\theta$}^{\prime})\right\\}$
$\displaystyle=$ $\displaystyle 0,$ (4.3b)
$\displaystyle\left\\{Q_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$}),\Psi^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{--}$}}(\mbox{\boldmath$\theta$}^{\prime})\right\\}$
$\displaystyle=$
$\displaystyle\delta^{(2)}(\mbox{\boldmath$\theta$},\mbox{\boldmath$\theta$}^{\prime})-\frac{1}{4\pi}.$
(4.3c)

One can now attempt to make an argument for the decoupling of hard and soft
degrees of freedom based on the following theorem in symplectic geometry,
which is proved in Appendix H:

Theorem: Let $(M,\Omega_{ab})$ be a symplectic manifold of finite dimension
$n$, with associated Poisson bracket defined by
$\left\\{f,g\right\\}=\Omega^{ab}\nabla_{a}f\nabla_{b}g$ for any functions
$f$, $g$ on $M$, where $\Omega^{ab}\Omega_{bc}=\delta^{a}_{\ c}$. Let ${\cal
S}_{2}:M\to M$ be a symplectomorphism. Suppose that there exists a set of
functions $s^{A}:M\to{\bf R}$ for $1\leq A\leq t$ which satisfies the two
properties:

  1. 1.

The functions are preserved under the symplectomorphism,

$s^{A}\circ{\cal S}_{2}=s^{A}.$ (4.4)

  2. 2.

The Poisson brackets

$\omega^{AB}=\left\\{s^{A},s^{B}\right\\}$ (4.5)

are constants on $M$ and form an invertible $t\times t$ matrix.

Then locally the symplectomorphism factorizes, that is, locally one can find
coordinates $(s^{1},\ldots,s^{t},h^{1},\ldots h^{n-t})$ for which the
symplectomorphism takes the form

$s^{A}\to s^{A},\ \ \ \ h^{\Gamma}\to{\bar{h}^{\Gamma}}(h^{\Sigma}),$ (4.6)

for $1\leq A\leq t$ and $1\leq\Gamma,\Sigma\leq n-t$.

Interpreting $s^{A}$ as soft and $h^{\Gamma}$ as hard degrees of freedom, the
theorem would seem to give sufficient conditions for the two sets of degrees
of freedom to decouple from one another. Indeed, if we take $s^{A}$ to be the
conserved quantities
$Q_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$})$ and
$\Psi^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{--}$}}(\mbox{\boldmath$\theta$})$
discussed above, and use the definition (1.4) of the scattering map ${\cal
S}_{2}$, then both conditions of the theorem are satisfied, from Eqs. (4.3).
It should follow that there exists a complementary set of hard variables that
are completely decoupled from the soft dynamics, as in Eq. (4.6). This is
essentially the decoupling argument of BP151515BP present their argument in
somewhat different terms: instead of invoking the theorem they attempt to
directly derive the change of phase space coordinates to the decoupled set
satisfying Eq. (4.6). They also focus on the gravitational case rather than
the electromagnetic case. However the theorem captures the essential idea of
their argument..

This argument fails for a subtle reason: the theorem is valid in finite
dimensions but not in the infinite dimensional context in which we want to
apply it. We can demonstrate this failure as follows. If the theorem were
applicable, it would establish the existence of a set of coordinates on phase
space which satisfy the following three conditions: (i) The coordinates are
independent; (ii) They are symplectically orthogonal (canonical); and (iii)
They include the conserved charges
$Q_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$})$ and
$\Psi^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{--}$}}(\mbox{\boldmath$\theta$})$.
Such coordinates do not in fact exist. To see this, we can start from the
coordinates (3.43) which satisfy (i) and (ii) but not (iii). Assuming that
coordinates exist which satisfy (i), (ii) and (iii), it follows that there
exists a symplectomorphism that maps $\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ onto
$Q_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ and ${\hat{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ onto $\Psi^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{--}$}}$. We attempt to
solve for this symplectomorphism by using a type II generating functional
following BP. This yields the following coordinates that satisfy (ii) and
(iii):

$\displaystyle{\tilde{\Psi}}^{\rm e}_{\rm new}(u,\mbox{\boldmath$\theta$})$
$\displaystyle=$ $\displaystyle{\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$})+g(u)\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$}),$ (4.7a)
$\displaystyle{\Psi}^{\rm m}_{\rm new}(u,\mbox{\boldmath$\theta$})$
$\displaystyle=$ $\displaystyle{\Psi}^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$}),$
(4.7b) $\displaystyle\chi_{\rm new}(u,\mbox{\boldmath$\theta$})$
$\displaystyle=$ $\displaystyle\exp\left\\{-i\left[{\hat{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$})-\frac{1}{2}\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$})-2\int
d{\bar{u}}g^{\prime}({\bar{u}}){\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}({\bar{u}},\mbox{\boldmath$\theta$})\right]\right\\}\,\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$}),$
(4.7c) $\displaystyle{\Delta\Psi}^{\rm e}_{\rm new}(\mbox{\boldmath$\theta$})$
$\displaystyle=$ $\displaystyle{\Delta\Psi}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$})+e^{2}D^{-2}Q^{\rm
hard}(\mbox{\boldmath$\theta$})[\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}],$
(4.7d) $\displaystyle{\hat{\Psi}}^{\rm e}_{\rm new}(\mbox{\boldmath$\theta$})$
$\displaystyle=$ $\displaystyle{\hat{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$})-\frac{1}{2}\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$})-2\int
dug^{\prime}(u){\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$}),$
(4.7e)

where $Q^{\rm hard}$ is defined by Eqs. (3.26) and (2.9a). One can check that
these new coordinates have the same Poisson brackets as the original $+$
coordinates, and so are symplectically orthogonal. However these coordinates
do not satisfy (i), i.e. they are not independent, since we have

$\Delta\Psi^{\rm e}_{\rm
new}(\mbox{\boldmath$\theta$})=2\lim_{u\to\infty}{\tilde{\Psi}}^{\rm e}_{\rm
new}(u,\mbox{\boldmath$\theta$})+e^{2}D^{-2}Q^{\rm
hard}(\mbox{\boldmath$\theta$})[\chi_{\rm new}],$ (4.8)

from Eqs. (2.46) and (2.48). What has happened is that the original
${\tilde{\Psi}}$ coordinate satisfied the boundary conditions
${\tilde{\Psi}}(u)\to 0$ as $u\to\pm\infty$, whereas the new ${\tilde{\Psi}}$
coordinate does not and so effectively encodes more information161616 A
similar phenomenon arises if one tries to diagonalize the Hamiltonian
functional  $H=\int du\int d^{2}\Omega(\partial_{u}\Psi)^{2}.$ (4.9) In finite
dimensions one can always find canonical coordinates that diagonalize a
positive definite quadratic Hamiltonian (Williamson’s theorem [63, 64]).
However Williamson’s theorem does not apply in our infinite dimensional
context: applying standard methods to find the canonical phase space
coordinates that diagonalize the Hamiltonian (4.9) gives rise to coordinates
that are not independent, just as in Eq. (4.8). The origin of the difficulty
in both cases is that the inverse (2.52) of the symplectic form (2.50) does
not lie in $\Gamma\otimes\Gamma$ as it would if the phase space $\Gamma$ were
finite dimensional, instead it lies in a larger space. See Ashtekar [41] for a
rigorous treatment of $\Gamma$ as a Fréchet space.. Thus the requirements of
an autonomous soft sector, independence of phase space coordinates, and
symplectic orthogonality of hard and soft sectors are incompatible.

A useful perspective on the dynamics can be obtained by defining an enlarged
phase space ${\tilde{\Gamma}}$ in which the would-be symplectomorphism (4.7)
is actually a symplectomorphism. This space is parameterized by the
coordinates on the left hand sides of Eqs. (4.7), where we demand that

$\lim_{u\to\infty}{\tilde{\Psi}}^{\rm e}_{\rm
new}(u,\mbox{\boldmath$\theta$})=-\lim_{u\to-\infty}{\tilde{\Psi}}^{\rm
e}_{\rm new}(u,\mbox{\boldmath$\theta$}),$ (4.10)

but we do not demand that this limit satisfy the constraint (4.8). The
enlarged phase space factors as ${\tilde{\Gamma}}={\tilde{\Gamma}}_{\rm
hard}\times{\tilde{\Gamma}}_{\rm soft}$, where ${\tilde{\Gamma}}_{\rm hard}$
is parameterized by the coordinates (4.7a) – (4.7c), and
${\tilde{\Gamma}}_{\rm soft}$ by the coordinates (4.7d) and (4.7e). We can
define a scattering map ${\tilde{\cal
S}}_{2}:{\tilde{\Gamma}}\to{\tilde{\Gamma}}$, which factorizes, acts on
${\tilde{\Gamma}}_{\rm soft}$ as the identity, and acts on
${\tilde{\Gamma}}_{\rm hard}$ as determined by the generating functional
(3.42) [note that the arguments of the generating functional ${\mathscr{G}}$
coincide with the coordinates (4.7a) – (4.7c)]. The physical phase space
$\Gamma$ is then given by restricting to the subspace of ${\tilde{\Gamma}}$
given by the constraint (4.8), and the physical scattering map ${\cal S}_{2}$
is given by restricting the action of ${\tilde{\cal S}}_{2}$ to this subspace.
Thus, although the scattering map does not factorize as argued by BP, it
descends from a mapping on a larger phase space which does.

In the enlarged phase space, even though the scattering map ${\tilde{\cal
S}}_{2}$ factorizes, the map on the hard factor ${\tilde{\Gamma}}_{\rm hard}$
is constrained by the existence of the soft conservation laws. In particular,
the right hand side of Eq. (4.8) is preserved, because of the the constraint
(G.7) on the generating functional ${\mathscr{G}}$. This is necessary for the
mapping ${\tilde{\cal S}}_{2}$ to preserve the physical phase space $\Gamma$.
Thus, the argument of BP for the triviality of the soft conservation laws —
that they act only on a soft sector and do not affect the dynamics in a hard
sector — does not apply.

###  4.3 Argument for decoupling based on factorization theorem for S matrix
elements

Another kind of argument for the triviality of scattering of soft degrees of
freedom was given by Mirbabayi and Porrati (henceforth MP) [32], based on the
soft theorems for S-matrix elements [2]. MP assumed that

  1. 1.

The system including soft degrees of freedom can be described by a Hilbert
space ${\cal H}$.

  2. 2.

There is a factorization of the Hilbert space into hard and soft factors

${\cal H}={\cal H}_{\rm hard}\otimes{\cal H}_{\rm soft},$ (4.11)

a basis $\left|a\right>$ of ${\cal H}_{\rm hard}$, and a set of unitary
operators $\Omega(a)$ on ${\cal H}_{\rm soft}$ for which the $S$ matrix can be
expressed as

$S=S_{ab}\left|a\right>\left<b\right|\Omega(a)\Omega(b)^{\dagger}.$ (4.12)

MP defined the factorization (4.11) in terms of an infrared cutoff on mode
energy, and took the basis $\left|a\right>$ of hard states to be the usual
particle number eigenstates used in Feynman diagrams. They then argued for the
decomposition (4.12) from soft theorems [2], with the operators $\Omega(a)$
providing the universal soft factors for $S$ matrix elements when some of the
ingoing and outgoing particles become soft.

The assumptions (4.11) and (4.12) are equivalent to a decoupling of two
sectors. We can write

$S=U(S_{\rm hard}\otimes 1)U^{\dagger}$ (4.13)

with $U$ being the unitary operator
$U=\sum_{a}\Omega(a)\left|a\right>\left<a\right|$ and $S_{\rm
hard}=S_{ab}\left|a\right>\left<b\right|$. It follows that there exists
another factorization of the Hilbert space ${\cal H}={\bar{\cal H}}_{\rm
hard}\otimes{\bar{\cal H}}_{\rm soft}$ for which the scattering matrix $S$
factorizes and acts as the identity on ${\bar{\cal H}}_{\rm soft}$. Taking the
classical limit it then follows that the scattering map ${\cal S}_{2}$ is of
the factorized form (4.6) discussed earlier, for some choice of hard and soft
variables.

This conclusion is in conflict with explicit computations within the classical
theory detailed in Secs. 5 and 6 below (where we use an exact decomposition
into hard and soft degrees of freedom rather than a decomposition based on
letting a cutoff go to zero). We conclude that the assumptions 1 and 2 above
are not valid in such a context: the correct description of the soft theorems
within the quantum theory must be more involved than Eqs. (4.11) and (4.12).
It is possible that no Hilbert space description of the dynamics exists when
soft degrees of freedom are included, as argued by Prabhu, Satishchandran, and
Wald [43] (see Sec. 7 below for further discussion). Another possibility is
that the Hilbert space cannot be expressed as a tensor product of hard and
soft factors, the appropriate structure might be something like the fusion
product defined by Donnelly and Freidel[49].

##  5 Explicit computation of scattering map: second order

###  5.1 Preamble

In the previous section we refuted some general arguments for the existence of
decoupled hard and soft sectors. However to show that the sectors are actually
always coupled requires an explicit computation of the dynamics. Such an
explicit computation in the context of perturbation theory is the goal of the
rest of this paper. We start in this section by considering the second order
scattering map, and proceed to third and fourth orders in the following
section.

###  5.2 Second order dynamics

The scattering at second order is trivial in the sense that all the
functionals defined in Eq. (3.34), that parameterize the nonlinear
interactions in preferred asymptotic gauge, vanish to this order:

${\cal H}^{\rm e}=O(\alpha^{3}),\ \ \ {\cal H}^{\rm
e}_{\infty}=O(\alpha^{3}),\ \ \ {\cal H}^{\rm m}=O(\alpha^{3}),\ \ \ {\cal
K}=O(\alpha^{3}).$ (5.1)

As a consequence, the explicit form of the scattering map coincides with the
linear order scattering map (3.23),

$\displaystyle\mathcal{P}_{\ast}\tilde{\Psi}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ $\displaystyle=$
$\displaystyle\tilde{\Psi}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}+O(\alpha^{3}),$ (5.2a)
$\displaystyle\mathcal{P}_{\ast}\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ $\displaystyle=$
$\displaystyle-\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle-$}}+O(\alpha^{3}),$ (5.2b)
$\displaystyle\mathcal{P}_{\ast}\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}$
$\displaystyle=$
$\displaystyle-\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}+O(\alpha^{3}),$
(5.2c) $\displaystyle\mathcal{P}_{\ast}\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ $\displaystyle=$
$\displaystyle\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}+O(\alpha^{3}),$ (5.2d)
$\displaystyle\mathcal{P}_{\ast}{\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ $\displaystyle=$
$\displaystyle{\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}+\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}+O(\alpha^{3}),$ (5.2e)

from Eqs. (2.47) and (3.29). In particular, the hard sector
$(\tilde{\Psi}^{\rm e},\Psi^{\rm m},\chi)$ and the soft sector
$(\Delta\Psi^{\rm e},{\bar{\Psi}}^{\rm e})$ are decoupled from one another and
evolve independently. Thus we will need go to higher order in perturbation
theory to find the leading order couplings between the soft and hard sectors.
We will return to this story in Sec. 6 below. In the remainder of this section
we will discuss some properties of the second order scattering and its
derivation.

Although the scattering is trivial in preferred asymptotic gauge, it is not
trivial in asymptotic Lorenz gauge, where there is a nonlinear soft-hard
interaction at quadratic order. This can be seen by substituting the result
(5.1) into the general Lorenz gauge scattering map (3.38), which yields from
Eqs. (2.47) and (3.29) that

$\displaystyle\mathcal{P}_{\ast}\tilde{{\underline{\Psi}}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ $\displaystyle=$
$\displaystyle\tilde{{\underline{\Psi}}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}+O(\alpha^{3}),$ (5.3a)
$\displaystyle\mathcal{P}_{\ast}{\underline{\Psi}}^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ $\displaystyle=$
$\displaystyle-{\underline{\Psi}}^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle-$}}+O(\alpha^{3}),$ (5.3b)
$\displaystyle\mathcal{P}_{\ast}{\underline{\chi}}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$
$\displaystyle=$
$\displaystyle-{\underline{\chi}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}+2i\Delta{\underline{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}{\underline{\chi}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}+O(\alpha^{3}),$
(5.3c) $\displaystyle\mathcal{P}_{\ast}\Delta{\underline{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ $\displaystyle=$
$\displaystyle\Delta{\underline{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}+O(\alpha^{3}),$ (5.3d)
$\displaystyle\mathcal{P}_{\ast}{\bar{\underline{\Psi}}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ $\displaystyle=$
$\displaystyle{\bar{\underline{\Psi}}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}-\Delta{\underline{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}+O(\alpha^{3}).$ (5.3e)

The nonlinear term in Eq. (5.3c) arises from the phase factor in Eq. (3.38),
and can be written in terms of the expansion coefficients (3.14) and (3.15) as

$\mathcal{P}_{\ast}{\underline{\chi}}^{(2)}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$})=2i\Delta{\underline{\Psi}}^{{\rm
e}(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\mbox{\boldmath$\theta$})\,{\underline{\chi}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(u,\mbox{\boldmath$\theta$}).$
(5.4)

The nonlinear scattering map (5.3), which is equivalent to the result (5.1),
is derived in Appendix I.

The nonlinearity in the second order Lorenz gauge scattering map (5.3) arises
only because of the presence of soft degrees of freedom. It vanishes if we
specialize to incoming data that is purely hard
($\Delta{\underline{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}={\bar{\underline{\Psi}}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}=0$). This is a well-known result,
familiar in the quantum context from the Feynman rules for scalar QED [65].
The argument is that quadratic scattering corresponds to a three particle
interaction and to the operator $A^{a}\Phi^{*}\nabla_{a}\Phi+$ c.c. in the
Lagrangian. If we combine the conservation of four-momentum and the fact that
the final momentum must be null to be on-shell, this forces all the momenta to
be collinear. Finally the Lorenz gauge condition forces the final result to
vanish. In Appendix I we translate this argument into the language of
perturbative classical scattering used in this paper, and explain how it
breaks down in the presence of soft degrees of freedom.

One consequence of the nonlinear soft-hard interaction is that the scattering
map (5.3) is not continuous, in the following sense. Consider a sequence
${}^{(n)}{\underline{\Psi}}^{\rm e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$
of incoming configurations, each of which has no soft part
[${}^{(n)}\Delta{\underline{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}={}^{(n)}{\bar{\underline{\Psi}}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}=0$], which converges pointwise to
${\underline{\Psi}}^{\rm e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$:

$\lim_{n\to\infty}{}^{(n)}{\underline{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(v,\mbox{\boldmath$\theta$})={\underline{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(v,\mbox{\boldmath$\theta$}).$ (5.5)

For each element in the sequence, the result (5.4) vanishes, and so the
$n\to\infty$ limit of the scattered fields also has this property. However the
result (5.4) does not vanish for the scattering of the $n\to\infty$ pointwise
limit (5.5) of the initial data. Thus, the scattering of soft degrees of
freedom cannot be obtained by simply taking a naive limit of hard scattering.
Equivalently, the scattering map is not determined by continuity by its
restriction to initial data that consists of smooth wavepacket states. This
property is specific to asymptotic Lorenz gauge, and does not occur in
preferred asymptotic gauge where the scattering map is continuous, as can be
seen from Eqs. (G.10) and (G.11) of Appendix G.

##  6 Explicit computation of scattering map: third and fourth orders

###  6.1 Overview

In the previous section we showed that the classical scattering map factorizes
into decoupled hard and soft sectors at quadratic order. We now proceed to
third and fourth orders in perturbation theory. We will show in this section
that no factorization into decoupled sectors is possible at these orders.

It will be sufficient for our arguments to consider only two specific pieces
of the fourth order scattering. The first piece is the change in
electromagnetic memory between $\mathscr{I}^{-}$ and $\mathscr{I}^{+}$. In
Sec. 6.2 and Appendix J we compute this quantity explicitly and show that it
is nonzero, as would be expected on general grounds [1]. The second piece is
the contribution to $\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ that is
linear in ${\bar{\Psi}}^{\rm e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ and
cubic in $\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}$, which we show in Sec.
6.2 is nonzero.

Next, in order to show that the hard and soft sectors are always dynamically
coupled, we need to address an ambiguity in the definitions of these sectors.
Roughly speaking the soft sector consists of zero energy modes, while the hard
sector consists of finite energy modes, but these requirements allow for
considerable leeway in the definitions. Our strategy will be to show that for
any definitions of hard and soft sectors, the two sectors are dynamically
coupled. However, we do require that the same definitions of hard and soft be
used at $\mathscr{I}^{-}$ and at $\mathscr{I}^{+}$ (otherwise it is always
trivially possible to find definitions which decouple the dynamics). As
discussed in the introduction, we adopt two different conventions for
identifying the phase spaces $\Gamma_{-}$ at $\mathscr{I}^{-}$ and
$\Gamma_{+}$ at $\mathscr{I}^{+}$, that allow us to enforce this requirement.

Section 6.3 considers the first convention, based on identifying
$\mathscr{I}^{-}$ and $\mathscr{I}^{+}$ using null geodesics. First, in Sec.
6.3.1, we consider cubic and quartic terms in the scattering map which couple
together the soft and hard sectors. Some of these terms are invariant under
linear field redefinitions and perturbative field redefinitions that alter the
definitions of the two sectors. We will call such terms invariant
interactions, as they cannot be removed by field redefinitions, and we derive
criteria for interactions to be invariant. Then, in Sec. 6.3.2, we show that
the two pieces of the fourth order scattering discussed in Sec. 6.2 are such
invariant interactions. Since these quantities are generically nonzero, the
hard and soft sectors are always coupled. These invariant interactions also
disallow scattering maps of the forms (1.5) and (1.6).

In Sec. 6.4 we turn to the second convention, which is based on the free field
scattering map. For this case there exist definitions of the soft variables
which are conserved, discussed in Sec. 4.2 above, and it is natural to
restrict to definitions of soft variables which preserve this property. With
this assumption we show that it is not possible to find definitions of soft
and hard variables for which the scattering map has the uncoupled form (1.1),
again using the results of Sec. 6.2.

###  6.2 Two nonzero pieces of fourth order scattering

In this section we focus attention on two particular pieces of the fourth
order scattering which we will show are nonzero.

The first piece is the total change $\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}-\mathcal{P}_{\ast}\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ in electromagnetic memory between
$\mathscr{I}^{-}$ and $\mathscr{I}^{+}$. From the conservation law (3.24) this
quantity is proportional to

$\Delta Q^{\rm hard}(\mbox{\boldmath$\theta$})=Q^{\rm
hard}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$})-\mathcal{P}_{\ast}Q^{\rm
hard}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\mbox{\boldmath$\theta$}),$
(6.1)

the change in the total hard charge per unit angle in the scalar field. It
also corresponds to the the functional ${\cal H}^{\rm e}_{\infty}$ in the
general parameterization (3.34) of the scattering map, from Eq. (3.36). It is
clear that this quantity is nonzero in general [1]. For example, for
electromagnetism coupled to massive charged particles, if one has two incoming
charged particles, there will in general be a nontrivial scattering and the
outgoing particles will have different directions of propagation from the
incoming ones. In this paper our source is a massless scalar field and so we
deal instead with wave packet ingoing and outgoing states, but it is clear
that generically there will be nontrivial scattering. In Appendix J we verify
this explicitly by computing the leading order contribution to $\Delta Q^{\rm
hard}(\mbox{\boldmath$\theta$})$, which is quartic in the initial scalar field
$\chi^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ and is given by the
expressions (J.4) and (J.6).

The second piece of the fourth order scattering that we consider is the
contribution to
$\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$})$
that is linear in ${\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ and cubic in
$\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}(v,\mbox{\boldmath$\theta$})$. We
can show that this contribution is nonzero as follows. Combining the leading
order formula (3.41) for the scattering map ${\cal S}_{2}$ in terms of the
generating functional $G$ together with the Poisson bracket (2.52d) gives that
the change in
$\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$})^{*}$
is

$-\frac{1}{2}\int du^{\prime}K(u-u^{\prime})\frac{\delta
G}{\delta\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u^{\prime},\mbox{\boldmath$\theta$})},$
(6.2)

where $K(u)=\Theta(u)-1/2$. Combining this with the specific form (3.42) of
the generating functional, the definition (1.4) of the scattering map ${\cal
S}_{2}$ and the free field scattering map (3.23) we obtain

$\mathcal{P}_{\ast}\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}=-\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}-\frac{1}{2}\int
du^{\prime}K(u-u^{\prime})e^{i{\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}}e^{i\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}/2}\frac{\delta{\mathscr{G}}}{\delta\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}}\left[{\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}+g\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}},-\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle-$}},-e^{-i{\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}}e^{-i\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}/2}\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}\right]^{*}.$
(6.3)

If we now specialize to a vanishing incoming electromagnetic field we obtain

$\mathcal{P}_{\ast}\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}=-\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}-\frac{1}{2}\int
du^{\prime}K(u-u^{\prime})\frac{\delta{\mathscr{G}}}{\delta\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}}\left[0,0,-\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}\right]^{*}.$
(6.4)

The second term here is computed explicitly in the last paragraph of Appendix
J and is cubic in $\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ and nonzero.
Consider now the contribution to the general result (6.3) that is linear in
${\bar{\Psi}}^{\rm e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ evaluated at
${\Delta\Psi}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}={\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}=0$. Because of the form of the
expression (6.3), this contribution can be computed from the known expression
(6.4) by inserting the complex exponential factors. Therefore we conclude that
the contribution which is linear in ${\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ and cubic in
$\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ is nonzero. We write this
contribution as

$\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}={\hat{F}}[{\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}},\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}},\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}},\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}]$
(6.5)

where the function ${\hat{F}}$ is linear in its first, second and third
arguments and antilinear in its fourth (see Appendix J).

###  6.3 Hard and soft sectors are coupled for first version of scattering map

We now specialize to the first version (1.3) of the scattering maps discussed
in the introduction. For this version the phase spaces $\Gamma_{-}$ and
$\Gamma_{+}$ are identified using the mapping $\varphi_{0}$ from
$\mathscr{I}^{-}$ to $\mathscr{I}_{+}$ defined with null geodesics, allowing
us to define the scattering map ${\cal S}_{1}$ from $\Gamma_{+}$ to
$\Gamma_{+}$. We will allow general definitions of hard and soft sectors of
$\Gamma_{+}$, and will not require these sectors to be symplectically
orthogonal.

####  6.3.1 Definition of invariant interactions

The general form (3.34) of the scattering map can be written schematically as

$\bar{y}^{a}=L^{a}_{\ b}y^{b}+M^{a}_{\ bcd}y^{b}y^{c}y^{d}+N^{a}_{\
bcde}y^{b}y^{c}y^{d}y^{e}+O(y^{5}),$ (6.6)

where $y^{a}$ are abstract phase space coordinates and the map is
parameterized in terms of some phase space tensors $L^{a}_{\ b}$, $M^{a}_{\
bcd}$ and $N^{a}_{\ bcde}$. Here we are using a notation where the indices
$a,b,\ldots$ run over the various fields and also encode the dependence of
these fields on the coordinates $u,\theta^{A}$, so that contractions over
these indices encompass integrals over these variables. The barred coordinates
${\bar{y}}^{a}$ refer to the final data on $\mathscr{I}^{+}$,

${\bar{y}}^{a}=\left({\tilde{\Psi}}_{\scalebox{0.7}{$\scriptscriptstyle+$}}^{\rm
e},\Psi_{\scalebox{0.7}{$\scriptscriptstyle+$}}^{\rm
m},\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Delta\Psi_{\scalebox{0.7}{$\scriptscriptstyle+$}}^{e},{\bar{\Psi}}_{\scalebox{0.7}{$\scriptscriptstyle+$}}^{\rm
e}\right),$ (6.7)

while the unbarred coordinates $y^{a}$ refer to the data on $\mathscr{I}^{+}$
obtained from the identification map (1.2) acting on the initial data on
$\mathscr{I}^{-}$. This mapping amounts to evaluating the initial data at
$v=u$ and applying the pullback $\mathcal{P}_{\ast}$, so the functions on
$\mathscr{I}^{+}$ are

$y^{a}=\mathcal{P}_{\ast}\left({\tilde{\Psi}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}^{\rm
e},\Psi_{\scalebox{0.7}{$\scriptscriptstyle-$}}^{\rm
m},\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}},\Delta\Psi_{\scalebox{0.7}{$\scriptscriptstyle-$}}^{e},{\bar{\Psi}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}^{\rm
e}\right).$ (6.8)

Finally the schematic scattering map (6.6) encodes the fact that there are no
quadratic terms when using preferred asymptotic gauge, cf. Eqs. (5.2) above,
so the leading nonlinearities arise at cubic order.

As in the previous section the phase space coordinates can be decomposed into
hard and soft components,

$y^{a}=(h^{A},s^{\Gamma}),$ (6.9)

where the hard variables $h^{A}$ refer to the first three fields in (6.8),
which depend on $u$ and $\theta^{A}$, while the soft variables $s^{\Gamma}$
refer to the last two fields in (6.8), which depend only on $\theta^{A}$. As
discussed in Sec. 3.4, the two sectors are uncoupled in the linear order
scattering map (3.23), so the tensor $L^{a}_{\ b}$ is block diagonal with
vanishing off-diagonal blocks:

$L^{\Gamma}_{\ A}=0,\ \ \ L^{A}_{\ \,\Gamma}=0.$ (6.10)

The diagonal block $L^{A}_{\ B}$ in the hard sector is given by Eqs. (3.23b) –
(3.23c), with the pullback operators $\mathcal{P}_{\ast}$ removed because of
Eq. (6.8), while the diagonal block $L^{\Gamma}_{\ \Sigma}$ in the soft sector
is similarly given by Eqs. (3.23d) and (3.23e) with $\mathcal{P}_{\ast}$
removed.

As we showed in Sec. 6.2 above, the hard and soft sectors are coupled via the
higher order terms in Eq. (6.6) which mix the two sectors together. Our goal
here is to determine when such interactions can be removed by field
redefinitions that alter the definitions of the hard and soft sectors. We
first consider perturbative field redefinitions of the form

$y^{a}=z^{a}+\Upsilon^{a}_{\ bcd}z^{b}z^{c}z^{d}+\Xi^{a}_{\
bcde}z^{b}z^{c}z^{d}z^{e}+O(z^{5}),$ (6.11)

which defines new phase space coordinates $z^{a}$, together with an identical
transformation for the barred variables. Here we have assumed that the
transformation is the identity to linear order, as linear transformations are
considered separately below. We also exclude any quadratic terms in the
transformation (6.11), since such terms would not generally maintain the
property of the scattering map (6.6) of having no quadratic terms171717 Terms
of the form $\rho^{a}_{\ bc}z^{b}z^{c}$ in Eq. (6.11) with $\rho^{a}_{\ bc}$
chosen to satisfy  $L^{a}_{\ e}\rho^{e}_{\ bc}=\rho^{a}_{\ ef}L^{e}_{\
b}L^{f}_{\ c}$ (6.12) will maintain the property of no quadratic terms in the
scattering map. However the contributions of such terms to Eqs. (6.15) vanish
when $\Upsilon^{a}_{\ bcd}=\Xi^{a}_{\ bcde}=0$, except for the contribution
$3M^{a}_{\ f(bc}\rho^{f}_{\ de)}-2\rho^{a}_{\ fg}M^{f}_{\ (bcd}L^{g}_{\ e)}$
(6.13) to Eq. (6.15c). These terms do not contribute to the interaction
(6.23). This follows from the fact that the condition (6.12) together with Eq.
(3.23) allow the following components of $\rho$ to be freely specified:
$\rho^{1}_{\ 11},\rho^{1}_{\ 14},\rho^{1}_{\ 12},\rho^{1}_{\ 33},\rho^{1}_{\
44},\rho^{1}_{\ 42},\rho^{1}_{\ 22},\rho^{3}_{\ 13},\rho^{3}_{\
34},\rho^{3}_{\ 32},\rho^{4}_{\ 14},\rho^{4}_{\ 44},\rho^{4}_{\
42},\rho^{5}_{\ 11},\rho^{5}_{\ 14},\rho^{5}_{\ 12},\rho^{5}_{\
33},\rho^{5}_{\ 44},\rho^{5}_{\ 42},\rho^{5}_{\ 22},\rho^{5}_{\
11},\rho^{5}_{\ 14}$, $\rho^{2}_{\ 12},\rho^{2}_{\ 33},\rho^{2}_{\
44},\rho^{2}_{\ 42},\rho^{2}_{\ 22}$, with the remaining components
constrained to vanish except for $\rho^{5}_{\ 15}=\rho^{4}_{\ 14}$,
$\rho^{5}_{\ 45}=\rho^{4}_{\ 44}/2$, $\rho^{5}_{\ 52}=\rho^{4}_{\ 42}$ and
those determined by $\rho^{a}_{\ bc}=\rho^{a}_{\ cb}$. Here we are using the
convention (6.8) for ordering the fields. Also the components $M^{4}_{\
133}=M^{{\Delta\Psi}^{\rm e}}_{{\tilde{\Psi}}^{\rm e}\chi\chi}$ and $M^{4}_{\
533}=M^{{\Delta\Psi}^{\rm e}}_{{\bar{\Psi}}^{\rm e}\chi\chi}$ in Eq. (6.13)
vanish from Eq. (I.28) and an equation analogous to Eq. (J.8) at one lower
order. Since general transformations can be obtained by composing those with
$\rho^{a}_{\ bc}=0$ and those with $\Upsilon^{a}_{\ bcd}=\Xi^{a}_{\ bcde}=0$,
we can without loss of generality neglect those with nonzero $\rho^{a}_{\
bc}$..

Using the transformation (6.11) and its inverse, we can write the scattering
map (6.6) in terms of the new phase space variables $z^{a}$. The result is

$\bar{z}^{a}={\hat{L}}^{a}_{\ b}z^{b}+{\hat{M}}^{a}_{\
bcd}z^{b}z^{c}z^{d}+{\hat{N}}^{a}_{\ bcde}z^{b}z^{c}z^{d}z^{e}+O(z^{5}),$
(6.14)

where the transformed tensors are

$\displaystyle{\hat{L}}^{a}_{\ b}$ $\displaystyle=$ $\displaystyle L^{a}_{\
b},$ (6.15a) $\displaystyle{\hat{M}}^{a}_{\ bcd}$ $\displaystyle=$
$\displaystyle M^{a}_{\ bcd}+L^{a}_{\ e}\Upsilon^{e}_{\ bcd}-\Upsilon^{a}_{\
efg}L^{e}_{\ b}L^{f}_{\ c}L^{g}_{\ d},$ (6.15b) $\displaystyle{\hat{N}}^{a}_{\
bcde}$ $\displaystyle=$ $\displaystyle N^{a}_{\ bcde}+L^{a}_{\ f}\Xi^{f}_{\
bcde}-\Xi^{a}_{\ fghi}L^{f}_{\ b}L^{g}_{\ c}L^{h}_{\ d}L^{i}_{\ e}.$ (6.15c)

We denote by ${\cal V}$ the linear space of tensors $(M^{a}_{\ bcd},N^{a}_{\
bcde})$. Consider now linear maps $\ell:{\cal V}\to{\bf R}$, elements of the
dual space ${\cal V}^{*}$. For example $\ell(M,N)$ could be a particular
component of the tensor $N$. We define ${\cal W}_{\rm mixed}$ to be the
subspace of ${\cal V}^{*}$ that is spanned by components of $M$ and $N$ with
both hard and soft indices, excluding the purely soft components
$(M^{\Sigma}_{\ \,\Gamma\Delta\Upsilon},N^{\Sigma}_{\
\,\Gamma\Delta\Upsilon\Lambda})$ and purely hard components $(M^{A}_{\
\,BCD},N^{A}_{\ \,BCDE})$. We will call maps $\ell$ in ${\cal W}_{\rm mixed}$
interactions, since they couple the hard and soft sectors together in the
dynamics. An example of such an interaction is the change in electromagnetic
memory discussed in Sec. 6.2 above. We define a linear map
${\mathfrak{A}}:{\cal V}\to{\cal V}$ that takes

${\mathfrak{A}}:(\Upsilon^{a}_{\ bcd},\Xi^{a}_{\ bcde})\to\left(L^{a}_{\
e}\Upsilon^{e}_{\ bcd}-\Upsilon^{a}_{\ efg}L^{e}_{\ b}L^{f}_{\ c}L^{g}_{\ d}\
,\ L^{a}_{\ f}\Xi^{f}_{\ bcde}-\Xi^{a}_{\ fghi}L^{f}_{\ b}L^{g}_{\ c}L^{h}_{\
d}L^{i}_{\ e}\right).$ (6.16)

We define the subspace ${\cal W}_{\rm invariant}$ of ${\cal V}^{*}$ to be the
set of maps $\ell$ for which $\ell\circ\mathfrak{A}=0$, which is the kernel of
the transpose of ${\mathfrak{A}}$. The space ${\cal W}_{\rm invariant}$
depends on the linear order scattering map $L^{a}_{\ b}$, for example if
$L^{a}_{\ b}=\delta^{a}_{\ b}$ then it is the entire space ${\cal V}^{*}$. The
key property of this definition is that nonzero interactions in ${\cal W}_{\rm
invariant}\cap{\cal W}_{\rm mixed}$ cannot be set to zero using the field
redefinitions (6.11), from Eqs. (6.15) and (6.16).

Turn now to linear field redefinitions of the form

$y^{a}=\Omega^{a}_{\ b}z^{b}.$ (6.17)

The transformed scattering map is again of the form (6.14) with the
transformed tensors being

$\displaystyle{\hat{L}}^{a}_{\ b}$ $\displaystyle=$
$\displaystyle\left(\Omega^{-1}\right)^{a}_{\ c}L^{c}_{\ d}\Omega^{d}_{\ b},$
(6.18a) $\displaystyle{\hat{M}}^{a}_{\ bcd}$ $\displaystyle=$
$\displaystyle\left(\Omega^{-1}\right)^{a}_{\ e}M^{e}_{\ fgh}\Omega^{f}_{\
b}\Omega^{g}_{\ c}\Omega^{h}_{\ d},$ (6.18b) $\displaystyle{\hat{N}}^{a}_{\
bcde}$ $\displaystyle=$ $\displaystyle\left(\Omega^{-1}\right)^{a}_{\
f}N^{f}_{\ ghij}\Omega^{g}_{\ b}\Omega^{h}_{\ c}\Omega^{i}_{\ d}\Omega^{j}_{\
e}.$ (6.18c)

We restrict attention to transformations $\Omega^{a}_{\ b}$ which preserve the
decoupling (6.10) of the two sectors at linear order:

${\hat{L}}^{\Gamma}_{\ A}=0,\ \ \ {\hat{L}}^{A}_{\ \,\Gamma}=0.$ (6.19)

Without loss of generality for analyzing decoupling we can assume that

$\Omega^{\Gamma}_{\ \Sigma}=\delta^{\Gamma}_{\ \Sigma},\ \ \ \Omega^{A}_{\
B}=\delta^{A}_{\ B}.$ (6.20)

The conditions (6.19) are then equivalent to, from Eqs. (6.10) and (6.18a),

$\Omega^{A}_{\ \Sigma}L^{\Sigma}_{\ \Gamma}=L^{A}_{\ B}\Omega^{B}_{\ \Gamma},\
\ \ \ \ \Omega^{\Gamma}_{\ B}L^{B}_{\ A}=L^{\Gamma}_{\
\Sigma}\Omega^{\Sigma}_{\ A}.$ (6.21)

For each such map $\Omega^{a}_{\ b}$, we define the map
${\mathfrak{A}}_{\Omega}:{\cal V}\to{\cal V}$ by

${\mathfrak{A}}_{\Omega}:(M^{a}_{\ bcd},N^{a}_{\ bcde})\to\left(M^{a}_{\
bcd}-\Omega^{-1\,a}_{\ \ e}M^{e}_{\ fgh}\Omega^{f}_{\ b}\Omega^{g}_{\
c}\Omega^{h}_{\ d}\ ,\ N^{a}_{\ bcde}-\Omega^{-1\,a}_{\ \ f}N^{f}_{\
ghij}\Omega^{g}_{\ b}\Omega^{h}_{\ c}\Omega^{i}_{\ d}\Omega^{j}_{\ e}\right).$
(6.22)

We define the subspace ${\cal W}^{\prime}_{\rm invariant}$ of ${\cal V}^{*}$
to be the set of maps $\ell$ for which $\ell\circ\mathfrak{A}_{\Omega}=0$ for
all $\Omega^{a}_{\ b}$ satisfying the conditions (6.20) and (6.21).
Equivalently, ${\cal W}^{\prime}_{\rm invariant}$ is the intersection of the
kernels of the transposes of the maps ${\mathfrak{A}}_{\Omega}$. From Eqs.
(6.18) and (6.22) we see that nonzero interactions $\ell$ in ${\cal
W}^{\prime}_{\rm invariant}\cap{\cal W}_{\rm mixed}$ cannot be set to zero
using the field redefinitions (6.17).

We will refer to the interactions in ${\cal W}_{\rm invariant}\cap{\cal
W}_{\rm invariant}^{\prime}\cap{\cal W}_{\rm mixed}$ as invariant
interactions. These interactions are not altered by either type of
transformation, linear or nonlinear. Which specific cubic and quartic
interactions are invariant depends on the details of the free field scattering
map $L^{a}_{\ b}$.

####  6.3.2 Two invariant nonzero interactions at quartic order

We next show that the two quartic interactions discussed in Sec. 6.2 above are
invariant interactions.

Consider first the change (6.1) in electromagnetic memory, specialized to the
case when there is an incoming scalar field but no electromagnetic field. We
show in Appendix J that this quantity is of order $O(\alpha^{4})$, so it
contributes to the tensor $N^{a}_{\ bcde}$. Since the incoming scalar field
$\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ is assumed to have no soft part
[cf. Eq(̇2.18) above] the corresponding component of $N$ is

$\ell(M,N)=N^{\Lambda}_{\ \,ABCD}$ (6.23)

where the $\Lambda$ index corresponds to the field $\Delta\Psi^{\rm e}$, and
the indices $A$, $B$, $C$ and $D$ correspond to the field $\chi$, from Eqs.
(3.34b), (3.35) and (6.9).

We now consider the action of the map (6.16) on the interaction (6.23). From
Eqs. (6.10), (6.16) and (6.23) we have that $\ell\circ{\mathfrak{A}}=0$ if

$L^{\Lambda}_{\ \Sigma}\Xi^{\Sigma}_{\ ABCD}=\Xi^{\Lambda}_{\ EFGH}L^{E}_{\
A}L^{F}_{\ B}L^{G}_{\ C}L^{H}_{\ D},$ (6.24)

where the indices $\Lambda$, $A$, $B$, $C$ and $D$ have the values discussed
after Eq. (6.23), and $\Xi^{a}_{\ bcde}$ is the transformation tensor defined
in Eq. (6.11). By using the linear order scattering map (3.23) together with
Eq. (6.8), we see that the effect of the mappings $L^{E}_{\ A}$ on the right
hand side is to replace $\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ with
$-\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}$. Similarly the mapping
$L^{\Lambda}_{\ \Sigma}$ on the left hand side acts as the identity map.
Therefore the condition (6.24) can be written as

$F[\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}]=F[-\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}],$
(6.25)

where we have defined the quartic functional
$F[\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}]$ to be $\Xi^{\Lambda}_{\
ABCD}y^{A}y^{B}y^{C}y^{D}$ with the same values of $\Lambda$ and $A,B,C,D$. On
the right hand side, the minus sign in the argument will cancel out, since the
term is a quartic function of this argument. Therefore the condition (6.25) is
satisfied, and so the interaction (6.23) is invariant under the perturbative
redefinitions (6.11) and is an element of ${\cal W}_{\rm invariant}\cap{\cal
W}_{\rm mixed}$.

We now turn to invariance under the linear phase space transformations (6.17).
These transformations are strongly constrained by the conditions (6.20) and
(6.21) and by the explicit form (3.23) of the free field scattering map. The
most general linear map consistent with these conditions is181818These
transformations encompass transformations caused by changes in the choice of
function $g(v)$ in the definition (2.47c), and also the change of variables
(2.51) used to diagonalize the Poisson brackets.

$\displaystyle{\tilde{\Psi}}^{\rm e}$ $\displaystyle\to$
$\displaystyle{\tilde{\Psi}}^{\rm e}+\beta[\Delta\Psi^{\rm e}],$ (6.26a)
$\displaystyle\Psi^{\rm m}$ $\displaystyle\to$ $\displaystyle\Psi^{\rm m},$
(6.26b) $\displaystyle\chi$ $\displaystyle\to$ $\displaystyle\chi,$ (6.26c)
$\displaystyle\Delta\Psi^{\rm e}$ $\displaystyle\to$
$\displaystyle\Delta\Psi^{\rm e},$ (6.26d) $\displaystyle{\bar{\Psi}}^{\rm e}$
$\displaystyle\to$ $\displaystyle{\bar{\Psi}}^{\rm
e}+\sigma[{\tilde{\Psi}}^{\rm e}],$ (6.26e)

where $\beta$ and $\sigma$ are linear functionals of their arguments. The
electromagnetic memory produced by an incoming scalar field with no incoming
electromagnetic field is invariant under these transformations. This follows
from the fact that it is the change in the quantity $\Delta\Psi^{\rm e}$,
which is invariant by Eq. (6.26d), and it is a functional only of $\chi$,
which is invariant by (6.26c).

Since the change in memory $\Delta Q^{\rm hard}(\mbox{\boldmath$\theta$})$ is
both generically nonzero from Sec. 6.2 and an invariant interaction, and since
it constitutes a transformation of incoming hard degrees of freedom to
outgoing soft degrees of freedom, we conclude that there is an unavoidable
coupling between the hard and soft sectors for the scattering map ${\cal
S}_{1}$. Furthermore this interaction rules out the form (1.6) of the
scattering map.

A similar argument applies to the interaction (6.5) and rules out the form
(1.5) of the scattering map. By paralleling the derivation of Eq. (6.25), we
find that this interaction will be invariant under the perturbative
redefinition (6.11) if

${\tilde{F}}[{\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}},\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}]=-{\tilde{F}}[{\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}},-\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}].$
(6.27)

Here we have defined the quartic functional ${\tilde{F}}[{\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}},\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}]$
to be $\Xi^{A}_{\Lambda BCD}y^{\Lambda}y^{B}y^{C}y^{D}$, where the index $A$
corresponds to the field $\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}$,
$\Lambda$ to the field ${\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ and $B$, $C$ and $D$ to the field
$\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}$. The condition (6.27) is
satisfied since ${\tilde{F}}$ is cubic in
$\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}$. Similarly one can check that
the interaction (6.5) is invariant under the linear field redefinition (6.18c)
using the explicit formula (6.26). Finally one can check that the conditions
of footnote 17 are satisfied using the conditions on $\rho^{a}_{\ bc}$ listed
there and the properties of third order scattering

$M^{\chi}_{\ {\bar{\Psi}}^{\rm e}{\bar{\Psi}}^{\rm e}\chi}=M^{\chi}_{\
{\tilde{\Psi}}^{\rm e}{\bar{\Psi}}^{\rm e}\chi}=M^{\Delta\Psi^{\rm e}}_{\
{\bar{\Psi}}^{\rm e}\chi\chi}=M^{\tilde{\Psi}^{\rm e}}_{\ {\bar{\Psi}}^{\rm
e}\chi\chi}=0,$ (6.28)

which can be derived using the methods of Appendices I and J.

###  6.4 Hard and soft sectors are coupled for second version of scattering
map

We next turn to the second version (1.4) of the scattering maps discussed in
the introduction. For this case there exist definitions of the soft variables
which are conserved, and it is natural to restrict to definitions of soft
variables which preserve this property. We will show that in this context it
is not possible to find definitions of hard and soft variables for which the
scattering map has the uncoupled form (1.1).

We take our initial definition of hard and soft sectors to be given by the
phase space coordinates

$y^{a}=(h^{A},s^{\Gamma})=\left({\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}},{\breve{\chi}}_{\scalebox{0.7}{$\scriptscriptstyle+$}},Q_{\scalebox{0.7}{$\scriptscriptstyle+$}},{\Psi}^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{--}$}}\right).$ (6.29)

Here the hard variables $h^{A}$ consist of the first three fields which depend
on $u$ and $\theta$, while the soft variables $s^{\Gamma}$ consist of the last
two fields, which depend only on $\theta$ and which are conserved in the
scattering as discussed in Sec. 4.2. Note that these phase space coordinates
are not symplectically orthogonal but they are independent, cf. the discussion
in Sec. 4.2 above. Also we have defined

${\breve{\chi}}_{\scalebox{0.7}{$\scriptscriptstyle+$}}=e^{-i\Psi^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{--}$}}}\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}},$
(6.30)

motivated by the form (3.42) of the generating functional; this definition
removes some (but not all) of the soft-hard coupling.

We denote by $\Sigma_{s}$ the subspace of the phase space
$\Gamma_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ given by fixing values of the
soft variables $s$. Taking the pullback of the symplectic form (2.50) to
$\Sigma_{s}$ yields a symplectic form $\Omega_{AB}$ whose inverse yields a
Dirac bracket on $\Sigma_{s}$ which we denote by $\left\\{\ ,\ \right\\}_{\rm
D}$. This Dirac bracket is easiest to compute starting from the Poisson
brackets of the variables (4.7). The result is that the Dirac brackets of the
hard variables $({\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}},{\breve{\chi}}_{\scalebox{0.7}{$\scriptscriptstyle+$}})$
coincide with the Poisson brackets (2.52) (with
$\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ replaced by
${\breve{\chi}}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$) except for the
bracket

$\left\\{D^{2}{\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$})\ ,\
{\breve{\chi}}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u^{\prime},\mbox{\boldmath$\theta$}^{\prime})\right\\}_{\rm
D}=ie^{2}g(u)\delta^{(2)}(\mbox{\boldmath$\theta$},\mbox{\boldmath$\theta$}^{\prime}){\breve{\chi}}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u^{\prime},\mbox{\boldmath$\theta$}^{\prime}).$
(6.31)

Since the soft variables $s$ are conserved, the restriction of the scattering
map ${\cal S}_{2}$ to $\Sigma_{s}$ gives a map ${\cal
S}_{s}:\Sigma_{s}\to\Sigma_{s}$, which we can write as

$h^{A}\to{\bar{h}}^{A}(h,s).$ (6.32)

When the right hand side is independent of $s$ then the two sectors are
uncoupled. If we make an $s$-dependent change of the coordinates $h^{A}$ on
$\Sigma_{s}$ (effectively changing the definition of the hard sector), the
effect is to make the replacement

${\cal S}_{s}\to{\bar{\cal S}}_{s}=\varphi_{s}\circ{\cal
S}_{s}\circ\varphi_{s}^{-1},$ (6.33)

for some diffeomorphism $\varphi_{s}:\Sigma_{s}\to\Sigma_{s}$. We do not
require that $\varphi_{s}$ be a symplectomorphism. We would like to show that
there is no choice of diffeomorphism that removes the $s$ dependence from the
right hand side of Eq. (6.32). We will show that attempting to derive such a
diffeomorphism runs into the same kind of obstacle as in Sec. 4.2 above: one
obtains a unique result but it is not a diffeomorphism, since the resulting
coordinates are not independent.

To show that no such diffeomorphism exists it is sufficient to work to linear
order in $s$ and to linear order in the deviation of ${\cal S}_{s}$ from the
identity map (appropriate to the small $h$ limit), since if it is impossible
to remove the coupling in this regime then it is impossible in general. Within
this limit we can write the mapping (6.32) as

$h^{A}\to h^{A}+\xi^{A}(h)+v^{A}(h,s),$ (6.34)

where $\xi^{A}$ is independent of $s$ and $v^{A}$ is linear in $s$. The effect
of the transformation (6.33) to linear order is to add to the right hand side
of Eq. (6.34) a term $({\cal L}_{\vec{\eta}}{\vec{\xi}})^{A}$ for some vector
field $\eta^{A}$ on $\Sigma_{s}$. Thus the condition for the dependence on $s$
to be removable is that there exists a $\eta^{A}$ which depends linearly on
$s$ for which

${\vec{v}}=-{\cal L}_{\vec{\eta}}{\vec{\xi}}.$ (6.35)

Next, the generating function (3.42) for the scattering map can be written in
terms of the coordinates (6.29) used in this section as

${\mathscr{G}}\left[{\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}+ge^{2}D^{-2}Q_{\scalebox{0.7}{$\scriptscriptstyle+$}}-ge^{2}D^{-2}Q^{\rm
hard}[{\breve{\chi}}_{\scalebox{0.7}{$\scriptscriptstyle+$}}],\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}},{\breve{\chi}}_{\scalebox{0.7}{$\scriptscriptstyle+$}}\right],$
(6.36)

from Eqs. (2.47), (3.24), (3.29) and (6.30). Note that this expression depends
on only one of the two soft variables,
$Q_{\scalebox{0.7}{$\scriptscriptstyle+$}}$. Evaluating this generating
functional at $Q_{\scalebox{0.7}{$\scriptscriptstyle+$}}=0$ and inserting into
the linearized transformation191919Using the Dirac brackets discussed around
Eq. (6.31). (3.41) yields the vector field $\xi^{A}$. Evaluating the piece of
${\mathscr{G}}$ linearized in $Q_{\scalebox{0.7}{$\scriptscriptstyle+$}}$,
which is

$e^{2}\int du\int
d^{2}\Omega\,g(u)\frac{\delta{\mathscr{G}}}{\delta{\breve{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$})}D^{-2}Q_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$}),$
(6.37)

gives instead the vector field $v^{A}$. Note that the quantity (6.37) is
generally nonzero, by Eqs. (3.33), (3.36), (G.6c) and the result of Sec. 6.2.
With these inputs we find that the solution to Eq. (6.35) for $\eta^{A}$ is

${\vec{\eta}}=-e^{2}\int du\int
d^{2}\Omega\,g(u)[D^{-2}Q_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$})]\frac{\delta}{\delta{\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$})}.$
(6.38)

However the corresponding coordinate transformation on $\Sigma_{s}$ is similar
to Eq. (4.7a), and leads to a set of coordinates which are not independent.
Thus, just as in Sec. 4.2, it is not possible to decouple the two sectors.

##  7 Discussion and conclusions

In this paper, we computed the perturbative classical scattering map from
$\mathscr{I}^{-}$ to $\mathscr{I}^{+}$ of electromagnetism coupled to a
massless, charged scalar field in four-dimensional Minkowski spacetime. We
consider various definitions hard and soft sectors of the theory, and adopt
two different conventions for how to ensure that the same definitions of hard
and soft are used at $\mathscr{I}^{-}$ and at $\mathscr{I}^{+}$. For both
conventions, we showed that the soft and hard sectors of the theory evolve
independently to quadratic order, but that at higher orders the two sectors
are coupled. In the first convention, this coupling between the hard and soft
sectors cannot be eliminated using linear or perturbative redefinitions of the
two sectors, which includes any possible “dressing” of the hard degrees of
freedom. In the second convention, the coupling cannot be eliminated by any
field redefinitions which preserve the fact that the soft variables are
conserved by the scattering process.

Our conclusions disagree with those of Ref. [33], which argued that there
exist definitions of the two sectors for which there is an exact decoupling.
We showed that the explanation for the disagreement is a property of
symplectic geometry which holds in finite dimensions but fails in infinite
dimensions.

We expect this nontrivial coupling of soft and hard sectors to generalize to
nonabelian gauge theories and to general relativity, and to the quantum as
well as classical regimes202020As discussed in Sec. 4.3, the statement that
the quantum theory does not admit a decomposition into two decoupled sectors
is not in conflict with the factorization theorems that constrain the soft
behavior of $S$ matrix elements [36, 37].. As a result, the infinitely many
conserved soft hair charges should yield non-trivial constraints on
gravitational scattering and on black hole formation and evaporation, as
argued in Refs. [23, 24].

To probe this issue further, it would be interesting to study the quantization
of the gauge theory studied in this paper, while carefully accounting for the
soft degrees of freedom. A recent detailed study of infrared issues in
scattering by Prabhu, Satishchandran and Wald (PSW) [43] shows that for
massless fields it is not possible to find in and out Hilbert spaces that
accommodate nontrivial changes in memory. However, as mentioned in Sec. 2.5
above, PSW start from an algebra of observables which is different from the
algebra obtained from our phase space $\Gamma$, due to our inclusion of the
edge mode (2.51) that is canonically conjugate to memory which PSW exclude as
being pure gauge. It would be interesting to study the consequences for the
analysis of PSW of adopting the different algebra used here.

## Acknowledgments

We thank Raphael Bousso, Mihalis Dafermos, Andrzej Herdegen, Shahin Sheikh
Jabbari, Mehrdad Mirbabayi, Kartik Prabhu, Gautam Satishchandran, Matthew
Schwartz and Bob Wald for helpful discussions and/or correspondence. We thank
an anonymous referee for some useful comments and suggestions. This work was
supported in part by NSF grants PHY-1707800 and PHY-2110463 to Cornell
University. IS was supported in part by the John and David Boochever prize
fellowship in fundamental theoretical physics. EF thanks the Department of
Applied Mathematics and Theoretical Physics at Cambridge University, where
part of this paper was written, for its hospitality.

##  Appendix A Symplectically orthogonal hard and soft sectors exclude certain
forms of scattering map

In this appendix we show that a scattering map ${\cal S}:\Gamma\to\Gamma$ on a
phase space $\Gamma$ cannot have the forms (1.5) or (1.6) if the soft and hard
variables are symplectically orthogonal. We assume a finite dimensional phase
space although we expect the result to generalize.

By assumption the phase space can be expressed as a product of hard and soft
factors $\Gamma=\Gamma_{\rm soft}\times\Gamma_{\rm hard}$, and we denote by
$y^{a}=(s^{A},h^{\Gamma})$ corresponding coordinates. Defining
$\Omega^{ab}=\\{y^{a},y^{b}\\}$, we have by assumption that

$0=\left\\{s^{A},h^{\Gamma}\right\\}=\Omega^{ab}\partial_{a}s^{A}\partial_{b}h^{\Gamma}.$
(A.1)

Applying the pullback operator ${\cal S}_{*}$ to this equation gives

$({\cal S}_{*}\Omega)^{ab}\partial_{a}({\cal S}_{*}s^{A})\partial_{b}({\cal
S}_{*}h^{\Gamma})=0.$ (A.2)

We now use the fact that the scattering map is a symplectomorphism so that
${\cal S}_{*}\Omega^{ab}=\Omega^{ab}$, and use the definitions
${\bar{s}}^{A}={\cal S}_{*}s^{A}=s^{A}\circ{\cal S}$,
${\bar{h}}^{\Gamma}={\cal S}_{*}h^{\Gamma}=h^{\Gamma}\circ{\cal S}$. This
gives $\Omega^{ab}\partial_{a}{\bar{s}}^{A}\partial_{b}{\bar{h}}^{\Gamma}=0$,
and expanding this out and using Eq. (A.1) again gives

$\Omega^{BC}\frac{\partial{\bar{s}}^{A}}{\partial
s^{B}}\frac{\partial{\bar{h}}^{\Gamma}}{\partial
s^{C}}=-\Omega^{\Sigma\Lambda}\frac{\partial{\bar{s}}^{A}}{\partial
h^{\Sigma}}\frac{\partial{\bar{h}}^{\Gamma}}{\partial h^{\Lambda}}.$ (A.3)

We now assume a scattering map of the form (1.5), which implies that

$\frac{\partial{\bar{h}}^{\Gamma}}{\partial s^{C}}=0,$ (A.4)

so the left hand side of Eq. (A.3) vanishes. On the right hand side the first
factor is invertible by Eq. (A.1) since $\Omega^{ab}$ is invertible, and
similarly the third factor is invertible by Eq. (A.4). This implies that

$\frac{\partial{\bar{s}}^{A}}{\partial h^{\Sigma}}=0,$ (A.5)

which contradicts the assumed form (1.5). A similar argument excludes the form
(1.6).

##  Appendix B Free Lorenz gauge solutions with nontrivial soft charges

In this appendix we review the solutions of the free, noninteracting theory
which satisfy everywhere the Lorenz gauge condition. We show that these
solutions satisfy the gauge-invariant asymptotic properties at $i^{+}$ and
$i^{-}$ given in Eqs. (2.21) and (2.22), but fail to satisfy the matching
condition (2.39) [cf. Eq. (B.11) below]. We then generalize the solutions to a
larger class which satisfy Lorenz gauge only asymptotically, and deduce the
general form of the Lorenz gauge scattering map for free solutions.

###  B.1 Global Lorenz gauge

In inertial coordinates $(t,x^{i})$ the homogeneous global Lorenz gauge
solutions are

$\displaystyle{\underline{A}}_{0}$ $\displaystyle=$ $\displaystyle 0,$ (B.1a)
$\displaystyle{\underline{A}}_{i}$ $\displaystyle=$ $\displaystyle\sum_{l\geq
0}\left\\{\partial_{ijL}\left[\frac{D_{jL}(t-r)}{r}-\frac{D_{jL}(t+r)}{r}\right]-\partial_{L}\left[\frac{D_{iL}^{\prime\prime}(t-r)}{r}-\frac{D_{iL}^{\prime\prime}(t+r)}{r}\right]\right\\}$
(B.1b) $\displaystyle+\sum_{l\geq
0}\epsilon_{ipq}\partial_{pL}\left[\frac{C_{qL}(t-r)}{r}-\frac{C_{qL}(t+r)}{r}\right].$

The notation here follows Ref. [66] and is as follows. The symbol $L$ is the
multi-index $L=(i_{1},i_{2},\ldots i_{l})$, and $D_{iL}$, $C_{iL}$ are
Cartesian tensors which are symmetric and trace-free on all of their indices.
The symbol $\partial_{L}$ means $\partial_{i_{1}}\ldots\partial_{i_{l}}$. Here
and throughout underlines mean that the corresponding quantities are in Lorenz
gauge. Note that, for a given solution, the tensors $D_{iL}$ and $C_{iL}$ are
not unique, since the solution (B.1) is invariant under transformations of the
form

$D_{iL}(x)\to D_{iL}(x)+\delta D_{iL}(x),\ \ \ \ C_{iL}(x)\to C_{iL}(x)+\delta
C_{iL}(x),$ (B.2)

for $l\geq 0$, where $\delta D_{iL}$ and $\delta C_{iL}$ are a polynomials in
$x$ of degree $2l+2$.

We will restrict attention to solutions for which the asymptotic behavior of
the symmetric tracefree tensors $D_{iL}$ and $C_{iL}$ as $x\to\infty$ is given
by

$D_{iL}(x)={\tilde{D}}_{+iL}(x)+P_{+iL}(x),\ \ \ \
C_{iL}(x)={\tilde{C}}_{+iL}(x)+Q_{+iL}(x),$ (B.3)

where ${\tilde{D}}_{+iL},{\tilde{C}}_{+iL}$ go to $0$ as $x\to\infty$, and
$P_{+iL}$ and $Q_{+iL}$ are polynomials in $x$ of degree $l+2$ and $l+1$
respectively. Similarly as $x\to-\infty$ we require that

$D_{iL}(x)={\tilde{D}}_{-iL}(x)+P_{-iL}(x),\ \ \ \
C_{iL}(x)={\tilde{C}}_{-iL}(x)+Q_{-iL}(x),$ (B.4)

where ${\tilde{D}}_{-iL},{\tilde{C}}_{-iL}\to 0$ as $x\to-\infty$, and
$P_{-iL}$ and $Q_{-iL}$ are again polynomials in $x$ of degree $l+2$ and $l+1$
respectively. Because of the invariance property (B.2), the solution (B.1)
depends only on the differences $\Delta P_{iL}=P_{+iL}-P_{-iL}$ and $\Delta
Q_{iL}=Q_{+iL}-Q_{-iL}$ between these polynomials and not on $P_{\pm iL}$ or
$Q_{\pm iL}$ individually. The assumptions (B.3) and (B.4) are compatible with
the large $r$ field expansions (2.6) and (2.14) that we have assumed212121This
would no longer be true if the polynomials $P_{\pm iL}$ and $Q_{\pm iL}$ were
of higher degree than $l+2$ and $l+1$., and also yield solutions with finite
energy.

The coefficients of the large $r$ expansions of the fields near
$\mathscr{I}^{+}$ are given by ${\underline{\cal A}}_{+u}={\underline{\cal
A}}_{+r}=0$, and

${\underline{\cal
A}}_{+A}(u,\mbox{\boldmath$\theta$})=\sum_{l}\,(-1)^{l+1}n^{L}e^{i}_{A}\left[{\tilde{D}}_{+iL}^{(l+2)}(u)+\epsilon_{ipq}n^{p}{\tilde{C}}_{+qL}^{(l+1)}(u)\right],$
(B.5)

where $n^{i}=x^{i}/r$, $e^{i}_{A}=D_{A}n^{i}$, $n^{L}=n^{i_{1}}\ldots
n^{i_{l}}$, and the superscripts $(l+2)$ and $(l+1)$ indicate the number of
derivatives taken. This expression satisfies the condition

${\underline{\cal A}}_{+A}\to 0$ (B.6)

as $u\to\infty$ at $i^{+}$, which yields the condition (2.22c), from Eq.
(2.7d). In the other limit $u\to-\infty$ at $i^{0}$ we have, from Eqs. (B.3)
and (B.4),

${\underline{\cal A}}_{+A}(u,\mbox{\boldmath$\theta$})\to{\underline{\cal
A}}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{--}$}A}(\mbox{\boldmath$\theta$})=\sum_{l}\,(-1)^{l}n^{L}e^{i}_{A}\left[\Delta
P_{iL}^{(l+2)}+\epsilon_{ipq}n^{p}\Delta Q_{qL}^{(l+1)}\right].$ (B.7)

Here the right hand side is independent of $u$ since $\Delta P_{iL}$ is a
polynomial of order $l+2$ and $\Delta Q_{qL}$ is a polynomial of order $l+1$.

Similar results apply for the limiting behavior of the solutions near
$\mathscr{I}^{-}$. The expansion coefficients are ${\underline{\cal
A}}_{-v}={\underline{\cal A}}_{-r}=0$, and

${\underline{\cal
A}}_{-A}(v,\mbox{\boldmath$\theta$})=\sum_{l}\,n^{L}e^{i}_{A}\left[{\tilde{D}}_{-iL}^{(l+2)}(v)-\epsilon_{ipq}n^{p}{\tilde{C}}_{-qL}^{(l+1)}(v)\right].$
(B.8)

This satisfies the condition as $v\to-\infty$ at $i^{-}$

${\underline{\cal A}}_{-A}\to 0,$ (B.9)

yielding the condition (2.21c), while as $v\to\infty$ at $i^{0}$ we have, from
Eqs. (B.3) and (B.4)

${\underline{\cal A}}_{-A}(v,\mbox{\boldmath$\theta$})\to{\underline{\cal
A}}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{--}{+}$}A}(\mbox{\boldmath$\theta$})=\sum_{l}\,n^{L}e^{i}_{A}\left[\Delta
P_{iL}^{(l+2)}-\epsilon_{ipq}n^{p}\Delta Q_{qL}^{(l+1)}\right].$ (B.10)

One can check that these Lorenz gauge solutions (B.1) satisfy the matching
conditions (2.23a) and (2.23b) [or equivalently (2.38)] discussed in the body
of the paper. Note, however, that they do not satisfy the matching condition
(2.39), instead this relation is satisfied with a sign flip, since from Eqs.
(B.7) and (B.10) we have

${\underline{\cal
A}}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{--}$}\,A}=-{\cal
P}_{*}{\underline{\cal
A}}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{--}{+}$}\,A}.$ (B.11)

This is discussed further in Sec. 3.2 above.

By expanding the solution (B.1) to subleading order in $1/r$ near
$\mathscr{I}^{+}$ and $\mathscr{I}^{-}$ and reading off the subleading
coefficients ${\hat{\cal A}}_{+u}$, ${\hat{\cal A}}_{+A}$, ${\hat{\cal
A}}_{-v}$ and ${\hat{\cal A}}_{-A}$, one can verify the limiting behavior at
$i^{-}$ and $i^{+}$ given by Eqs. (2.21a), (2.21b), (2.22a) and (2.22b).
Similar analyses for free massless scalar field solutions establishes (2.21d)
and (2.22d).

###  B.2 Asymptotic Lorenz gauge

The solutions (B.1) are the most general free solutions that obey the Lorenz
gauge condition everywhere in spacetime. However one can obtain a more general
class of solutions in asymptotic Lorenz gauge, in which one imposes the Lorenz
gauge condition only at $r\geq R$ for some $R$. Specifically for $r\geq R$ one
can transform the solutions according to

${\underline{A}}_{a}\to{\underline{A}}_{a}+\nabla_{a}\varepsilon,\ \ \ \
{\underline{\Phi}}\to e^{i\varepsilon}{\underline{\Phi}},\ \ \ \
\varepsilon=\sum_{l\geq
1}D_{L}\partial_{L}\left(\frac{u^{l}+v^{l}}{r}\right),$ (B.12)

where $D_{L}$ for $l\geq 1$ are some constant traceless symmetric tensors. For
$r<R$ one can use any smooth extension of $\varepsilon$. This transformation
preserves the asymptotic gauge conditions (2.34), and the initial and final
data transform as ${\underline{\cal
A}}_{\scalebox{0.7}{$\scriptscriptstyle+$}A}\to{\underline{\cal
A}}_{\scalebox{0.7}{$\scriptscriptstyle+$}A}+D_{A}\varepsilon_{\scalebox{0.7}{$\scriptscriptstyle+$}}$,
${\underline{\cal
A}}_{\scalebox{0.7}{$\scriptscriptstyle-$}A}\to{\underline{\cal
A}}_{\scalebox{0.7}{$\scriptscriptstyle-$}A}+D_{A}\varepsilon_{\scalebox{0.7}{$\scriptscriptstyle-$}}$,
where
$\varepsilon_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\mbox{\boldmath$\theta$})$
is a freely specifiable function with only $l\geq 1$ components222222This
function is given explicitly in terms of the coefficients $D_{L}$ in Eq.
(B.12) by $\varepsilon_{\scalebox{0.7}{$\scriptscriptstyle+$}}=\sum_{l\geq
1}c_{l}D_{L}n^{L}$ with $c_{l}=2\sum_{i=0,\ i\ {\rm
even}}^{l}2(i-1)(i-3)\ldots(i-2l+1)$. and
$\varepsilon_{\scalebox{0.7}{$\scriptscriptstyle-$}}={\cal
P}_{*}\varepsilon_{\scalebox{0.7}{$\scriptscriptstyle+$}}$. The transformation
is of the even form (2.35), and thus is not gauge but instead is a mapping
from solutions to physically distinct solutions, as discussed in Sec. 2.4.

The general asymptotic Lorenz gauge solutions will no longer satisfy the
conditions (B.6) and (B.9) at past and future timelike infinity. However there
is a combination of these conditions which is unaffected by the transformation
(B.12), and which is still valid for the general solutions, namely

${\underline{\Psi}}^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{+}$}}={\cal
P}_{*}{\underline{\Psi}}^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{--}{--}$}}.$ (B.13)

###  B.3 Scattering map for free solutions

The scattering map that relates the initial data on $\mathscr{I}^{-}$ to the
final data on $\mathscr{I}^{+}$ for the free global Lorenz gauge solutions can
be read off from Eqs. (B.5) and (B.8) and their analogs for scalar fields, and
is

$\displaystyle{\underline{\chi}}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$})$
$\displaystyle=$ $\displaystyle-{\cal
P}_{*}\,{\underline{\chi}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(u,\mbox{\boldmath$\theta$}),$
(B.14a) $\displaystyle{\underline{\cal
A}}_{\scalebox{0.7}{$\scriptscriptstyle+$}A}(u,\mbox{\boldmath$\theta$})$
$\displaystyle=$ $\displaystyle{\cal P}_{*}\left[{\underline{\cal
A}}_{\scalebox{0.7}{$\scriptscriptstyle-$}A}(u,\mbox{\boldmath$\theta$})-{\underline{\cal
A}}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{--}{+}$}A}(\mbox{\boldmath$\theta$})\right].$
(B.14b)

This scattering map can be generalized to the class of asymptotic Lorenz gauge
solutions generated by the transformation (B.12) by expressing the gauge
transformation function $\varepsilon_{\scalebox{0.7}{$\scriptscriptstyle+$}}$
in terms of the new initial data. Writing the result in terms of the
potentials (2.36), making use of the vanishing magnetic charges condition
(2.41) and using ${\cal P}_{*}\varepsilon_{AB}=-\varepsilon_{AB}$ gives

$\displaystyle{\underline{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$})$
$\displaystyle=$ $\displaystyle{\cal P}_{*}\left[{\underline{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(u,\mbox{\boldmath$\theta$})-{\underline{\Psi}}^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{--}{+}$}}(\mbox{\boldmath$\theta$})+{\underline{\Psi}}^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{--}{--}$}}(\mbox{\boldmath$\theta$})\right],$
(B.15a) $\displaystyle{\underline{\Psi}}^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$})$
$\displaystyle=$ $\displaystyle-{\cal P}_{*}{\underline{\Psi}}^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(u,\mbox{\boldmath$\theta$}),$
(B.15b)
$\displaystyle{\underline{\chi}}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,\mbox{\boldmath$\theta$})$
$\displaystyle=$ $\displaystyle-{\cal
P}_{*}\,{\underline{\chi}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(u,\mbox{\boldmath$\theta$}).$
(B.15c)

##  Appendix C Properties of interacting Lorenz gauge solutions

In this appendix we deduce some properties of nonlinear Lorenz gauge
solutions. Consider first solutions in global Lorenz gauge. We claim that the
conditions (B.6) and (B.9) at future and past timelike infinity are still
satisfied by these solutions. To see this, consider the version of the theory
on the Einstein static universe obtained by making a conformal transformation.
In this version spacetime is compact and all the fields are bounded. Consider
now a small neighborhood ${\cal V}$ of future timelike infinity $i^{+}$. At
each order in perturbation theory, one can obtain the fields inside ${\cal V}$
by specifying the initial conditions on the Cauchy surface obtained by taking
the intersection of $\partial{\cal V}$ with the image of Minkowski spacetime
on the Einstein static universe cylinder. The fields inside ${\cal V}$ (and in
particular the limit to $i^{+}$ of the fields on $\mathscr{I}^{+}$) are given
as a sum of a homogeneous solution determined by the initial data, and an
inhomogeneous solution determined by the sources inside ${\cal V}$ with zero
initial data. However, we know from Appendix B that the homogeneous solutions
must satisfy the vanishing condition (B.6) at $i^{+}$, since they are free
Lorenz gauge solutions. The inhomogeneous solution can give a nonvanishing
contribution to the limit, however this can be made arbitrarily small by
taking the size of the neighborhood to zero, since the sources are bounded. We
conclude that the conditions (B.6) and (B.9) are satisfied.

Just as for the free solutions, more general nonlinear solutions in asymptotic
Lorenz gauge obtained from the transformation (B.12) will no longer satisfy
the conditions (B.6) and (B.9), but will satisfy the condition (B.13).

##  Appendix D Presymplectic form is independent of choice of Cauchy surface

In the body of the paper, we showed that the presymplectic form (2.26) is
independent of the choice of Cauchy surface $\Sigma$, for the cases
$\Sigma=\mathscr{I}^{-}$ and $\Sigma=\mathscr{I}^{+}$, and for field
perturbations of the special form (2.29) given the condition (2.39). In this
appendix we review and modify slightly the results of Campiglia and Eyheralde
(CE) [58], that give sufficient conditions for extending the independence of
Cauchy surface to arbitrary Cauchy surfaces and to arbitrary field
perturbations. A closely related treatment has been given by Henneaux and
Troessaert [67].

Consider first spacelike Cauchy surfaces $\Sigma$ that limit to spatial
infinity $i^{0}$. We denote by ${\cal H}$ the unit hyperboloid of the space of
directions of approach to the point of spatial infinity. This hyperboloid acts
like a spacetime boundary, and the presymplectic form $\Omega_{\Sigma}$ can
potentially depend on the Cauchy surface $\Sigma$ through the location of the
two-surface ${\cal S}$ where $\Sigma$ intersects ${\cal H}$. Under the
appropriate assumptions of the asymptotic fall offs of the fields near ${\cal
H}$ [Eq. (4.2) of CE)], one can compute the pullback $\theta$ of the
presymplectic potential of the theory to ${\cal H}$. This pullback can be
decomposed as

$\theta=-\delta\ell+d\beta+{\cal E},$ (D.1)

the sum of a total variation boundary term $-\delta\ell$, an exact corner term
$d\beta$, and a residual term ${\cal E}$ called the flux term. The general
theory for obtaining a presymplectic form that is independent of Cauchy
surface [58, 47] consists of (i) Modifying the definition (2.26) of the
presymplectic form to exclude the corner term,

$\Omega_{\Sigma}=\delta\int_{\Sigma}\theta-\delta\int_{\cal S}\beta;$ (D.2)

and (ii) Specializing the decomposition (D.1) and the definition of the field
configuration space to make the flux term vanish.

CE find a nonzero flux term in general, the second term in their Eq. (4.5).
This term can be eliminated by specializing the definition of the field
configuration space. CE impose an asymptotic Lorenz gauge condition, their Eq.
(4.8). However that Lorenz gauge condition is incompatible with our condition
(2.39), and instead yields Eq. (2.39) with a sign flip [cf. Eq. (B.11) above].
Therefore we instead impose a modified version of their Eq. (4.8) with the
first term omitted. That modified condition can always be enforced by making
an appropriate “gauge transformation” of the form of their Eq. (2.38),
although as explained after Eq. (2.39) above the transformation should not be
thought of as a gauge specialization, and instead should be thought of as a
specialization of the definition of the field configuration space. Our
modified version of their condition (4.8) eliminates the flux term and also
implies Eq. (2.39)232323This follows from the fact that, in their notation,
the even parity piece of $A^{0}_{\alpha}$ is of the form $D_{\alpha}\lambda$
for some $\lambda$ with $D_{\alpha}D^{\alpha}\lambda=0$, and their Eq. (3.2)..

CE also find a nonzero corner term proportional to the pullback to ${\cal H}$
of $\epsilon_{abcd}A^{[c}\delta A^{d]}$, their Eq. (4.6)242424Note that they
use different terminology and call it a boundary term, and they also work with
$\omega=\delta\theta$ rather than $\theta$.. With the corner term subtraction
included, the presymplectic form (D.2) is independent of the choice of
spacelike Cauchy surface $\Sigma$ that limits to $i^{0}$. The corner term on
${\cal H}$ vanishes when one takes the limit to the boundaries of ${\cal H}$,
using their Eqs. (4.8) [the modified version], (4.6), (3.9), (B.2), footnote
3, and assuming the specialization given in their footnote 9. It also vanishes
on null infinity, with the gauge specializations (2.34). It follows that the
corner term subtraction does not affect the presymplectic forms
$\Omega_{\mathscr{I}^{-}}$ and $\Omega_{\mathscr{I}^{+}}$, and that
$\Omega_{\mathscr{I}^{-}}$ and $\Omega_{\mathscr{I}^{+}}$ coincide with the
common value of $\Omega_{\Sigma}$ for all the spacelike Cauchy surfaces
$\Sigma$.

##  Appendix E Field configuration space in the magnetic sector

In this appendix we discuss a difference between the space of solutions for
the electric potential $\Psi_{\scalebox{0.7}{$\scriptscriptstyle-$}}^{\rm e}$
and the magnetic potential $\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$. For the class of Lorenz gauge
free solutions discussed in Appendix B these potentials vanish at $i^{-}$,
from Eq. (B.9):

${\underline{\Psi}}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{--}{--}$}}^{\rm
e}=0,\ \ \ \
{\underline{\Psi}}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{--}{--}$}}^{\rm
m}=0,$ (E.1)

and this generalizes to the solutions of the interacting theory. However, by
using the transformation freedom (2.35a) we can make
$\Psi_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{--}{--}$}}^{\rm e}$ be
nonzero, from Eq. (2.37a). Thus in the full space of solutions with our
preferred asymptotic gauge conditions (2.34) and (2.39) [which do not include
Lorenz gauge] we have that both
$\Psi_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{--}{--}$}}^{\rm e}$ and
$\Psi_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{--}{+}$}}^{\rm e}$ are
nonzero in general.

For the magnetic variables the story is somewhat different. Starting from the
class of solutions which satisfy (E.1), one can attempt to obtain a larger
class of solutions by analogy with the procedure for the electric variables,
by making a transformation of the initial data on $\mathscr{I}^{-}$ of the
form

$\Psi_{\scalebox{0.7}{$\scriptscriptstyle-$}}^{\rm
e}\to\Psi_{\scalebox{0.7}{$\scriptscriptstyle-$}}^{\rm e},\ \ \ \ \
\Psi_{\scalebox{0.7}{$\scriptscriptstyle-$}}^{\rm
m}\to\Psi_{\scalebox{0.7}{$\scriptscriptstyle-$}}^{\rm
m}+{\tilde{\varepsilon}}_{\scalebox{0.7}{$\scriptscriptstyle-$}},$ (E.2)

where ${\tilde{\varepsilon}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ is a
function on $\mathscr{I}^{-}$ with no $l=0$ component that is independent of
$v$. Although the transformation (E.2) is not a gauge transformation, it is a
kind of magnetic analog of the gauge transformation given by Eqs. (2.37) [57].
By solving the equations of motion one can determine the effects of this
transformation of the initial data on the entire solution, and in particular
the data on $\mathscr{I}^{+}$ transforms as [cf. Eq. (2.35a) above]

$\Psi_{\scalebox{0.7}{$\scriptscriptstyle+$}}^{\rm
e}\to\Psi_{\scalebox{0.7}{$\scriptscriptstyle+$}}^{\rm e},\ \ \ \ \
\Psi_{\scalebox{0.7}{$\scriptscriptstyle+$}}^{\rm
m}\to\Psi_{\scalebox{0.7}{$\scriptscriptstyle+$}}^{\rm
m}+{\tilde{\varepsilon}}_{\scalebox{0.7}{$\scriptscriptstyle+$}}.$ (E.3)

where ${\tilde{\varepsilon}}_{\scalebox{0.7}{$\scriptscriptstyle+$}}={\cal
P}_{*}{\tilde{\varepsilon}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$. The
magnetic charges

${\tilde{Q}}_{\tilde{\varepsilon}}=\frac{1}{e^{2}}\int
d^{2}\Omega{\tilde{\varepsilon}}_{\scalebox{0.7}{$\scriptscriptstyle+$}}{\cal
F}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{--}$}AB}=\frac{1}{e^{2}}\int
d^{2}\Omega{\tilde{\varepsilon}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}{\cal
F}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{--}{+}$}AB}$ (E.4)

associated with these transformations [cf. Eq. (2.23b) above] can be derived
in exact parallel with the derivation for the electric case discussed in Sec.
2.4 [57]. However, the solutions generated by the transformations (E.2) are
generically singular in the interior of the spacetime. One example of such a
solution is a static magnetic dipole at the origin $r=0$. In this paper we
restrict attention to initial data on $\mathscr{I}_{-}$ which evolves into
smooth solutions in the interior of the spacetime, which requires that

$\Psi_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{--}{--}$}}^{\rm m}=0,$
(E.5)

and disallows the transformations (E.2). A similar analysis at
$\mathscr{I}^{+}$ yields the condition

$\Psi_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{+}$}}^{\rm m}=0.$
(E.6)

The other limits
$\Psi_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{--}$}}^{\rm m}$ and
$\Psi_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{--}{+}$}}^{\rm m}$ of
the magnetic potentials at spatial infinity are generally nonzero (see
Appendix B). However in this paper we restrict attention to the sector of the
theory in which they vanish, as discussed around Eq. (2.41).

##  Appendix F Asymptotic edge modes

###  F.1 Equivalence of soft degrees of freedom and edge modes

In the literature on soft charges it has been conjectured [1] that soft modes
can be alternatively described in the language of edge modes used in Refs.
[49, 50, 51, 52]. In this appendix we demonstrate the equivalence explicitly
by giving an alternative construction of the phase space and symplectic
product of Sec. 2.5 using edge modes. A general treatment of edge modes in
electromagnetism at finite boundaries has been given by Freidel and Pranzetti
[53].

We slightly generalize the treatment in the body of the paper by considering a
region ${\cal R}$ of $\mathscr{I}^{-}$ of points with $v_{1}\leq v\leq v_{2}$.
At the end we will specialize to $v_{1}=-\infty$, $v_{2}=\infty$, the case
treated in the body of the paper. The presymplectic product evaluated on this
region is not gauge invariant: the derivation that led to Eq. (2.31b) gives in
this context

$\Omega_{{\cal R}}(\delta
A_{a},\delta\Phi;\delta_{\varepsilon}A_{a},\delta_{\varepsilon}\Phi)=\frac{1}{e^{2}}\int
d^{2}\Omega\left[\delta{\cal
F}_{\scalebox{0.7}{$\scriptscriptstyle-$}rv}\varepsilon_{\scalebox{0.7}{$\scriptscriptstyle-$}}\right]^{v_{2}}_{v_{1}}.$
(F.1)

Now the field component at the boundary $v=v_{1}$ is given by, from Eq.
(2.21a) and an equation analogous to Eq. (2.10a),

${\cal
F}_{\scalebox{0.7}{$\scriptscriptstyle-$}rv}(v_{1})=\int_{-\infty}^{v_{1}}dv\left[e^{2}{\cal
J}_{\scalebox{0.7}{$\scriptscriptstyle-$}v}-D^{A}{\cal
F}_{\scalebox{0.7}{$\scriptscriptstyle-$}Av}\right].$ (F.2)

If we now vary just the fields within the region ${\cal R}$, as is relevant
for the presymplectic form (F.1), the result vanishes, since the field ${\cal
F}_{\scalebox{0.7}{$\scriptscriptstyle-$}rv}$ at $v_{1}$ depends only on
fields outside the region:

$\delta{\cal F}_{\scalebox{0.7}{$\scriptscriptstyle-$}rv}(v_{1})=0.$ (F.3)

Hence the one of the two terms in (F.1) vanishes. However the second term at
$v=v_{2}$ is generally nonzero and so the presymplectic form is not gauge
invariant.

The key idea of edge modes is to use the Stueckelberg trick252525A nice review
of this technique and its applications in gauge theory as well as
gravitational theories can be found in Ref. [68]. to restore gauge invariance
by introducing new degrees of freedom [49, 50, 51]. We introduce a $U(1)$
field $e^{ie^{2}\lambda}$ on the boundary at $v=v_{2}$, which transforms under
gauge transformations as
$\lambda\to\lambda+\varepsilon_{\scalebox{0.7}{$\scriptscriptstyle-$}}/e^{2}$.
We add to the presymplectic form $\Omega_{\cal R}$ a term

$\int d^{2}\Omega\left[\delta_{1}\lambda\delta_{2}{\cal
F}_{\scalebox{0.7}{$\scriptscriptstyle-$}vr}(v_{2})-\delta_{2}\lambda\delta_{1}{\cal
F}_{\scalebox{0.7}{$\scriptscriptstyle-$}vr}(v_{2})\right],$ (F.4)

so that the total presymplectic form is now gauge invariant.

We next specialize the gauge by using the gauge function
$\varepsilon_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ to set ${\cal
A}_{\scalebox{0.7}{$\scriptscriptstyle-$}v}$ to zero, using the analog of Eq.
(2.12a). There is a residual gauge freedom given by

$\varepsilon_{\scalebox{0.7}{$\scriptscriptstyle-$}}=\varepsilon_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\mbox{\boldmath$\theta$}).$
(F.5)

We now follow the derivation given in the body of the paper that leads to the
form (2.50) of the presymplectic form, but keeping the extra term (F.4). The
final result is

$\displaystyle\Omega_{{\cal R}}$ $\displaystyle=$
$\displaystyle-\frac{1}{e^{2}}\int dv\int
d^{2}\Omega\left[\partial_{v}{\tilde{\Psi}}^{\rm
e}_{1\,\scalebox{0.7}{$\scriptscriptstyle-$}}D^{2}{\tilde{\Psi}}^{\rm
e}_{2\,\scalebox{0.7}{$\scriptscriptstyle-$}}+\partial_{v}\Psi^{\rm
m}_{1\,\scalebox{0.7}{$\scriptscriptstyle-$}}D^{2}\Psi^{\rm
m}_{2\,\scalebox{0.7}{$\scriptscriptstyle-$}}-(1\leftrightarrow 2)\right]$
(F.6) $\displaystyle-\frac{1}{e^{2}}\int
d^{2}\Omega\left\\{D^{2}\Delta\Psi^{\rm
e}_{1\,\scalebox{0.7}{$\scriptscriptstyle-$}}\left[{\bar{\Psi}}^{\rm
e}_{2\,\scalebox{0.7}{$\scriptscriptstyle-$}}-\lambda_{2}+2\int
dvg^{\prime}{\tilde{\Psi}}^{\rm
e}_{2\scalebox{0.7}{$\scriptscriptstyle-$}}\right]-(1\leftrightarrow
2)\right\\}$ $\displaystyle+\int dv\int
d^{2}\Omega\left[\partial_{v}\delta_{1}\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}^{*}\delta_{2}\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}+\partial_{v}\delta_{1}\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}\delta_{2}\chi_{\scalebox{0.7}{$\scriptscriptstyle-$}}^{*}-(1\leftrightarrow
2)\right].$

We may now use the residual gauge freedom (F.5) (which was not a gauge freedom
in the body of the paper) to set to zero the variable ${\bar{\Psi}}^{{\rm
e}}_{2\scalebox{0.7}{$\scriptscriptstyle-$}}$. Then the phase space and
presymplectic form (F.6) coincide exactly with the version (2.50) from the
body of the paper (when we specialize to $v_{1}=-\infty$, $v_{2}=\infty$),
except that the variable ${\bar{\Psi}}^{{\rm
e}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ has been replaced by $-\lambda$.
Thus we see that of the two canonically conjugate variables $\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ and ${\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ that comprise the soft modes, one
can be interpreted as an edge mode at the future boundary at $v=v_{2}=\infty$
of $\mathscr{I}^{-}$.

###  F.2 Necessity of including edge modes

As discussed in the introduction and in Sec. 2.5, there is some disagreement
in the literature on the necessity of including asymptotic edge modes.
Ashtekar [40, 41] eliminates these modes by imposing $\Psi^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{--}$}}=\Psi^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{--}{+}$}}=0$ in the
definition of the field configuration space. Bousso and collaborators argue in
the gravitational case that the edge modes are unobservable [54, 33]. The
issue arises in a different guise in the algebraic approach to the scattering
problem, where one focuses on constructing the algebra of observables. The
recent analysis of infrared scattering by Prabhu, Satishchandran and Wald [43]
is based on an algebra obtained by excluding the edge modes. However that
algebra could be modified to incorporate them262626Specifically the algebra of
observables constructed in their Sec. 4.2 could be extended by including the
additional quantity (2.51) (suitably smeared over angles), which satisfies the
requirement of their Eq. (2.5).. Finally, the edge modes are included in the
treatments by Strominger and collaborators [48, 1].

One argument for including edge modes is that they are required in order for
there to be an action on the phase space of the large gauge transformations
(2.35a) associated with the new conserved charges (2.40) [48, 1].

Another argument is that it is not possible to eliminate edge modes using a
gauge transformation (since the corresponding transformations are not
degeneracy directions of the presymplectic form and are thus not true gauge
transformations). However, they can be instead eliminated by specializing the
definition of the field configuration space.

The strongest argument for including edge modes is as follows. We will give
the argument for the gravitational case. The edge modes (or soft hair) in that
case are observable only relative to a choice of convention, a BMS frame,
equivalent to the choice of zero readings on a set of clocks on an asymptotic
two sphere. This is the reason that the edge modes are sometimes argued to be
unobservable [54, 33]. However, the same non-observability argument would also
apply to the direction in space of the angular momentum of a stationary
gravitational source, which is observable only relative to a choice of
convention. One could make a diffeomorphism to fix the direction of angular
momentum to be in the $z$ direction. However, quantum fluctuations in the
direction of angular momentum have physical consequences, and cannot be
correctly described in a phase space where angular momentum always points in
the $z$ direction. By analogy, one expects quantum fluctuations in
gravitational edge modes to have physical consequences, and they would not be
correctly described in a phase space in which the edge modes are fixed.

Finally we note that the results of this paper are unchanged if the edge modes
are excluded. This can be seen as follows. The general form of the scattering
map (3.34) transforms appropriately under the transformation freedom (3.27)
and (3.28). Eliminating the edge modes ${\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ and ${\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ fixes this transformation freedom
and also eliminates Eq. (3.34b). However the soft-hard coupling discussed in
Sec. 6 is still be present in the second term on the right hand side of Eq.
(3.34a), specifically in the difference between its $u\to\pm\infty$ limits. As
in Sec. 6 this soft-hard coupling cannot be eliminated.

##  Appendix G Derivation of generating functional representation of
scattering map

In this appendix we derive some properties of the generating functional $G$
for the scattering map which were discussed in Sec. 3.6.

First, in order for the scattering map ${\cal S}$ to be of the specific form
(3.34) derived in Sec. 3.5, the generating functional $G$ must be of the form

$G\left[{\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}},{\hat{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}\right]={\cal
G}\left[{\breve{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}},e^{-i{\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}}e^{i{\Delta\Psi}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}}\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}\right]$
(G.1)

for some functional ${\cal G}$, where ${\breve{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ and ${\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ are defined in terms of the fields
on the left hand side by Eqs. (2.51) and (3.29). In deriving the
representation (G.1) we used the decomposition (3.39) of the scattering map,
the generating functional parameterization (3.41), the Poisson brackets
(2.52), and the linear order scattering map (3.23).

We next define variational derivatives of the the functional ${\cal
G}[{\breve{\Psi}}^{\rm e},\Psi^{\rm m},\chi]$. These are defined by

$\displaystyle\delta{\cal G}$ $\displaystyle=$ $\displaystyle\int du\int
d^{2}\Omega\frac{\delta{\cal G}}{\delta\Psi^{\rm
m}(u,\mbox{\boldmath$\theta$})}\delta\Psi^{\rm
m}(u,\mbox{\boldmath$\theta$})+\int du\int d^{2}\Omega\left[\frac{\delta{\cal
G}}{\delta\chi(u,\mbox{\boldmath$\theta$})}\delta\chi(u,\mbox{\boldmath$\theta$})+{\rm
c.c.}\right]$ (G.2) $\displaystyle+\int du\int d^{2}\Omega\frac{\delta{\cal
G}}{\delta{\breve{\Psi}}^{\rm
e}(u,\mbox{\boldmath$\theta$})}\delta{\breve{\Psi}}^{\rm
e}(u,\mbox{\boldmath$\theta$})+\int d^{2}\Omega\frac{\delta{\cal
G}}{\delta{\Delta\Psi}^{\rm
e}(\mbox{\boldmath$\theta$})}\delta{\Delta\Psi}^{\rm
e}(\mbox{\boldmath$\theta$}).$

The only nontrivial point here is that the field variation

${\breve{\Psi}}^{\rm e}\to{\breve{\Psi}}^{\rm e}+\delta{\breve{\Psi}}^{\rm
e}=\breve{\Psi}^{\rm e}+\delta{\tilde{\Psi}}^{\rm e}+g(u)\delta\Delta\Psi^{\rm
e},$ (G.3)

generates two terms in the variation $\delta{\cal G}$ of the functional, a
bulk term [first term on the second line of Eq. (G.2)], and a boundary term
(second term on the second line), since the variation
$\delta{\breve{\Psi}}^{e}$ is generally nonzero at the boundaries
$u=\pm\infty$, with

$\delta{\breve{\Psi}}^{\rm e}(\pm\infty)=\pm\delta\Delta\Psi^{\rm e}/2.$ (G.4)

Also, the integral of $\delta{\cal G}/\delta{\breve{\Psi}}^{\rm
e}(u,\mbox{\boldmath$\theta$})$ against any test function
$f(u,\mbox{\boldmath$\theta$})$ is only defined when
$f(\infty,\mbox{\boldmath$\theta$})=-f(-\infty,\mbox{\boldmath$\theta$})$.

One might worry that there is some inconsistency in the definition (G.2) since
the variables ${\breve{\Psi}}^{\rm e}$ and $\Delta\Psi^{\rm e}$ are not
independent, by Eq. (G.4). However there is no inconsistency, as can be seen
by formulating the definitions in terms of the independent variables
${\tilde{\Psi}}^{\rm e}$ and $\Delta\Psi^{\rm e}$ and then translating using
Eq. (G.3), which yields

$\displaystyle\frac{\delta{\cal G}}{\delta{\breve{\Psi}}^{\rm
e}(u,\mbox{\boldmath$\theta$})}$ $\displaystyle=$
$\displaystyle\left(\frac{\delta{\cal G}}{\delta{\tilde{\Psi}}^{\rm
e}(u,\mbox{\boldmath$\theta$})}\right)_{\Delta\Psi^{\rm e}},$ (G.5a)
$\displaystyle\frac{\delta{\cal G}}{\delta{\Delta\Psi}^{\rm
e}(\mbox{\boldmath$\theta$})}$ $\displaystyle=$
$\displaystyle\left(\frac{\delta{\cal G}}{\delta{\Delta\Psi}^{\rm
e}(\mbox{\boldmath$\theta$})}\right)_{{\tilde{\Psi}}^{\rm e}}-\int
dug(u)\left(\frac{\delta{\cal G}}{\delta{\tilde{\Psi}}^{\rm
e}(u,\mbox{\boldmath$\theta$})}\right)_{{\Delta\Psi}^{\rm e}}.$ (G.5b)

We use the variational derivatives on the left hand side rather than those on
the right hand side, since they are independent of the choice of function
$g(u)$.

We can now compute the various functionals ${\cal H}^{\rm m}$, ${\cal K}$,
${\cal H}^{\rm e}$ and ${\cal H}^{\rm e}_{\infty}$ that appear in the
scattering map (3.34) in terms of the generating functional ${\cal G}$. By
using the generating functional parameterization (3.41) with $y^{a}$ taken to
be $\Psi^{\rm m}$, $\chi$, ${\tilde{\Psi}}^{\rm e}$ and $\Delta\Psi^{\rm e}$,
respectively, the Poisson brackets (2.52), the variational derivative
definitions (G.2), the decomposition (3.39) of the scattering map, the linear
order scattering map (3.23), and comparing with the definitions (3.34c),
(3.34d), (3.37a) and (3.35), we can read off the following expressions:

$\displaystyle{\cal H}^{\rm m}[u,\mbox{\boldmath$\theta$};{\breve{\Psi}}^{\rm
e},\Psi^{\rm m},\chi]$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{2}D^{-2}\int
du^{\prime}K(u-u^{\prime})\frac{\delta{\cal G}}{\delta\Psi^{\rm
m}(u^{\prime},\mbox{\boldmath$\theta$})}\left[{\breve{\Psi}}^{\rm
e},-\Psi^{\rm m},-\chi\right],\ \ \ $ (G.6a) $\displaystyle{\cal
K}[u,\mbox{\boldmath$\theta$};{\breve{\Psi}}^{\rm e},\Psi^{\rm m},\chi]$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\int
du^{\prime}K(u-u^{\prime})\frac{\delta{\cal
G}}{\delta\chi(u^{\prime},\mbox{\boldmath$\theta$})}\left[{\breve{\Psi}}^{\rm
e},-\Psi^{\rm m},-\chi\right]^{*},$ (G.6b) $\displaystyle{\cal H}^{\rm
e}[u,\mbox{\boldmath$\theta$};{\breve{\Psi}}^{\rm e},\Psi^{\rm m},\chi]$
$\displaystyle=$ $\displaystyle\frac{1}{2}e^{2}D^{-2}\int
du^{\prime}K(u-u^{\prime})\frac{\delta{\cal G}}{\delta{\breve{\Psi}}^{\rm
e}(u^{\prime},\mbox{\boldmath$\theta$})}\left[{\breve{\Psi}}^{\rm
e},-\Psi^{\rm m},-\chi\right],\ \ \ $ (G.6c) $\displaystyle{\cal H}^{\rm
e}_{\infty}[\mbox{\boldmath$\theta$};{\breve{\Psi}}^{\rm e},\Psi^{\rm
m},\chi]$ $\displaystyle=$ $\displaystyle\frac{i}{2}e^{2}D^{-2}\int
du\,\chi(u,\mbox{\boldmath$\theta$})\frac{\delta{\cal
G}}{\delta\chi(u,\mbox{\boldmath$\theta$})}\left[{\breve{\Psi}}^{\rm
e},-\Psi^{\rm m},-\chi\right]+{\rm c.c.},\ \ \ \ $ (G.6d)

where $K(u)=\Theta(u)-1/2$. These expressions are accurate up to corrections
of order $O({\cal G}^{2})$ or equivalently $O(\alpha^{6})$.

The generating functional ${\cal G}$ must obey two additional constraints.
First, by imposing consistency of the formulae (G.6c) and (G.6d) with the
definition (3.33), and making the substitutions $\Psi^{\rm m}\to-\Psi^{\rm m}$
and $\chi\to-\chi$, we obtain

$\int du\frac{\delta{\cal G}}{\delta{\breve{\Psi}}^{\rm
e}(u,\mbox{\boldmath$\theta$})}[{\breve{\Psi}}^{\rm e},\Psi^{\rm
m},\chi]=-2i\int du\left\\{\chi(u,\mbox{\boldmath$\theta$})\frac{\delta{\cal
G}}{\delta{\chi}(u,\mbox{\boldmath$\theta$})}[{\breve{\Psi}}^{\rm e},\Psi^{\rm
m},\chi]\right\\}+{\rm c.c.}+O({\cal G}^{2}).$ (G.7)

Second, using the generating functional parameterization (3.41) of the
scattering map with $y^{a}=\hat{\Psi}^{\rm e}$, making use of the Poisson
bracket (2.52c) and the variational derivative definitions (G.2), and
comparing with the scattering map (3.37b) and making use of the formulae
(G.6c) and (G.6d) for the functionals ${\cal H}^{\rm e}$ and ${\cal H}^{\rm
e}_{\infty}$, we obtain the constraint

$\frac{\delta{\cal G}}{\delta{\Delta\Psi}^{\rm
e}(\mbox{\boldmath$\theta$})}[{\breve{\Psi}}^{\rm e},\Psi^{\rm
m},\chi]=-\frac{i}{2}\int
du\left\\{\chi(u,\mbox{\boldmath$\theta$})\frac{\delta{\cal
G}}{\delta{\chi}(u,\mbox{\boldmath$\theta$})}[{\breve{\Psi}}^{\rm e},\Psi^{\rm
m},\chi]\right\\}+{\rm c.c.}.+O({\cal G}^{2}).$ (G.8)

By direct substitution one can show that the general solution of this equation
is

${\cal G}\left[{\breve{\Psi}}^{\rm e},\Psi^{\rm
m},\chi\right]={\mathscr{G}}\left[{\breve{\Psi}}^{\rm e},\Psi^{\rm
m},e^{-\frac{i}{2}\Delta\Psi^{\rm e}}\chi\right]+O(\alpha^{6}),$ (G.9)

for some functional ${\mathscr{G}}$, with the restriction that the boundary
variational derivative defined in Eq. (G.2) associated with the first argument
of ${\mathscr{G}}$ vanishes:

$\left(\frac{\delta{\mathscr{G}}}{\delta\Delta\Psi^{\rm e}}\right)_{\rm first\
argument}=0.$ (G.10)

Substituting the expression (G.9) into the definition (G.1) of ${\cal G}$
gives that the original generating functional $G$ can be written as

$G\left[{\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}},{\hat{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}\right]={\mathscr{G}}\left[{\breve{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}},\Psi^{\rm
m}_{\scalebox{0.7}{$\scriptscriptstyle+$}},e^{-i{\bar{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}}e^{i{\Delta\Psi}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}/2}\chi_{\scalebox{0.7}{$\scriptscriptstyle+$}}\right]+O(\alpha^{6}).$
(G.11)

The form of the generating functional given by Eqs. (G.10) and (G.11) is
actually valid to all orders in $\alpha$ [cf. Eq. (3.42) above], even though
we have derived it here only to $O(\alpha^{5})$. This is because (i) it
guarantees that the quantity $\Psi^{\rm
e}_{\scalebox{0.55}{$\scriptscriptstyle\stackanchor{+}{--}$}}={\hat{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}-2\int
dug^{\prime}{\tilde{\Psi}}^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}-\Delta\Psi^{\rm
e}_{\scalebox{0.7}{$\scriptscriptstyle+$}}/2$ is preserved under the
transformation (3.40), from the Poisson brackets (2.52) and Eqs. (3.39) and
(3.23), thus ensuring the matching condition (2.39); and (ii) it similarly
guarantees that the conservation law (3.36) is satisfied.

##  Appendix H Sufficient conditions for decoupling in finite dimensions

In this appendix we prove the theorem in symplectic geometry given in Sec.
4.2, which gives sufficient conditions for two sets of degrees of freedom to
decouple in a finite dimensional system.

The derivation is essentially an application of Frobenius’ theorem for vector
fields [69]. We define the symmetry generator (Hamiltonian) vector fields
${\vec{v}}^{(A)}$ associated with the conserved quantities $s^{A}$ by

$v^{(A)a}=-\Omega^{ab}\nabla_{b}s^{A}$ (H.1)

for $1\leq A\leq t$. In order for Frobenius’ theorem to be applicable, the Lie
brackets of these vector fields with each other must be expressible as linear
combinations of the vector fields. However, the Lie bracket
$\left[{\vec{v}}^{(A)},{\vec{v}}^{(B)}\right]$ is just the Hamiltonian vector
field associated with the Poisson bracket (4.5), which vanishes since
$\omega^{AB}$ are assumed to be constants, $\nabla_{a}\omega^{AB}=0$. Thus we
obtain $\left[{\vec{v}}^{(A)},{\vec{v}}^{(B)}\right]=0$.

From Frobenius’ theorem it now follows that the subbundle of the tangent
bundle over $M$ defined by the span of the vector fields ${\vec{v}}^{(A)}$ is
integrable. That is, locally there exist coordinates $(x^{A},h^{\Gamma})$ for
$1\leq A\leq t$ and $1\leq\Gamma\leq n-t$ for which the vector fields
${\vec{v}}^{(A)}$ are tangent to the surfaces of constant $h^{\Gamma}$:

${\vec{v}}^{(A)}=\chi^{AB}(x,h)\frac{\partial}{\partial x^{B}}$ (H.2)

for some invertible $\chi^{AB}$. However this implies that
$\omega^{AB}={\vec{v}}^{(A)}(s^{B})=\chi^{AC}\partial s^{B}/\partial x^{C}$,
and since $\omega^{AB}$ and $\chi^{AC}$ are both invertible it follows from
the inverse function theorem that at fixed $h$ we can we can express $x^{A}$
as a function of $s^{A}$. Hence without loss of generality we can take
$x^{A}=s^{A}$.

It now follows that in these coordinates $(s^{A},h^{\Gamma})$ we have
$\Omega^{AB}=\omega^{AB}$ and
$\Omega^{A\Gamma}=\left\\{s^{A},h^{\Gamma}\right\\}=-v^{(A)\Gamma}=0$ from Eq.
(H.2). Thus the symplectic form can be expressed as

$\Omega=\omega_{AB}ds^{A}\wedge
ds^{B}+\Omega_{\Gamma\Sigma}(s,h)dh^{\Gamma}\wedge dh^{\Sigma},$ (H.3)

where $\omega^{AB}\omega_{BC}=\delta^{A}_{\ C}$. From $d\Omega=0$ it follows
that $\Omega_{\Gamma\Sigma}$ is independent of $s$, and demanding that the
symplectic form (H.3) be preserved under the pull back ${\cal S}_{2*}$ now
shows that the symplectomorphism ${\cal S}_{2}$ must have the form (4.6).

##  Appendix I Computation of Lorenz gauge scattering map to second order

In this appendix we derive the second order Lorenz gauge scattering map (5.3).

Our starting point is the set of free Lorenz gauge solutions with soft charges
described in Appendix B. Rather than using the symmetric trace-free tensor
expansion used there, it will be convenient to represent those solutions as
conventional plane wave expansions, with the soft degrees of freedom encoded
in distributional components of the mode coefficients at zero frequency. We
write the first order scalar field as

${\underline{\Phi}}^{(1)}(t,{\bf x})=\sum_{\eta=\pm}\int d^{3}k\,e^{-i\eta
kt}e^{i{\bf k}\cdot{\bf x}}f_{\eta}({\bf k}),$ (I.1)

where $k=|{\bf k}|$ and the functions $f_{+}({\bf k})$ and $f_{-}({\bf k})$
are independent. These functions are related to the initial data
${\underline{\chi}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(v,{\bf
n})=\lim_{r\to\infty}r{\underline{\Phi}}^{(1)}(v-r,r{\bf n})$ on
$\mathscr{I}^{-}$, with ${\bf n}={\bf x}/r$, by [cf. Eq. (9.0.29) of Ref. [1]]

$f_{\eta}({\bf k})=-\frac{i\eta}{2\pi
k}{\tilde{\underline{\chi}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\eta
k,-\eta{\hat{\bf k}}).$ (I.2)

Here $\eta=\pm 1$, ${\hat{\bf k}}={\bf k}/k$ and
${\tilde{\underline{\chi}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ is
the Fourier transform

${\tilde{\underline{\chi}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\omega,{\bf
n})=\frac{1}{2\pi}\int dve^{i\omega
v}{\underline{\chi}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(v,{\bf
n}).$ (I.3)

Similarly the first order vector potential is given by
${\underline{A}}^{(1)t}=0$ and

${\underline{{\bf A}}}^{(1)}(t,{\bf x})=\sum_{\gamma=\pm}\int
d^{3}p\,e^{-i\gamma pt}e^{i{\bf p}\cdot{\bf x}}{\bf a}_{\gamma}({\bf p}),$
(I.4)

with $p=|{\bf p}|$,

${\bf p}\cdot{\bf a}_{\gamma}({\bf p})=0$ (I.5)

and ${\bf a}_{\gamma}(-{\bf p})^{*}={\bf a}_{-\gamma}({\bf p})$. The mode
coefficients ${\bf a}_{\gamma}({\bf p})$ are related to the initial data
${\underline{\cal A}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}A}(v,{\bf
n})$ defined in Sec. 2.1 by

$a^{i}_{\gamma}({\bf p})=-\frac{i\gamma}{2\pi p}{\tilde{\underline{\cal
A}}}^{(1)i}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\gamma p,-\gamma{\hat{\bf
p}}).$ (I.6)

Here $\gamma=\pm 1$, ${\hat{\bf p}}={\bf p}/p$, ${\tilde{\underline{\cal
A}}}^{(1)i}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\omega,{\bf
n})={\tilde{\underline{\cal
A}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}A}(\omega,{\bf
n})h^{AB}e_{B}^{i}$, $\theta^{A}=(\theta,\varphi)$ are coordinates on the two-
sphere, $h_{AB}$ is the metric given by
$h_{AB}d\theta^{A}d\theta^{B}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$,
$h^{AB}h_{BC}=\delta^{A}_{\ C}$, $e_{A}^{i}=\partial n^{i}/\partial\theta^{A}$
and

${\tilde{\underline{\cal
A}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}A}(\omega,{\bf
n})=\frac{1}{2\pi}\int dve^{i\omega v}{\underline{\cal
A}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}A}(v,{\bf n}).$ (I.7)

We next discuss how the soft charges are encoded in the mode coefficients. For
the incoming scalar field, we have excluded by assumption any soft degrees of
freedom, and the fall off assumption (2.18) implies that the function
${\tilde{\underline{\chi}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\omega,{\bf
n})$ is bounded. For the incoming vector vector potential, on the other hand,
using the fall-off assumption (2.17), the definitions (2.47) and (2.36b), and
the vanishing magnetic charges condition (2.41) we obtain for the Fourier
transform

${\tilde{\underline{\cal
A}}}^{(1)i}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\omega,{\bf
n})=\frac{i}{2\pi}\Delta{\underline{\cal
A}}^{(1)i}_{\scalebox{0.7}{$\scriptscriptstyle-$}}({\bf n})\,{\rm
P.V.}\frac{1}{\omega}+{\overline{\underline{\cal
A}}}^{(1)i}_{\scalebox{0.7}{$\scriptscriptstyle-$}}({\bf
n})\delta(\omega)+{\tilde{\underline{\cal A}}}^{(1)i\,{\rm
rest}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\omega,{\bf n}),$ (I.8)

where P.V. means “principal value”,

$\Delta{\underline{\cal
A}}^{(1)i}_{\scalebox{0.7}{$\scriptscriptstyle-$}}({\bf
n})=e^{i}_{A}h^{AB}D_{B}\Delta{\underline{\Psi}}^{{\rm
e}\,(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}({\bf n}),\ \ \ \
{\overline{\underline{\cal
A}}}^{(1)i}_{\scalebox{0.7}{$\scriptscriptstyle-$}}({\bf
n})=e^{i}_{A}h^{AB}D_{B}{\bar{\underline{\Psi}}}^{{\rm
e}\,(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}({\bf n}),$ (I.9)

and

${\tilde{\underline{\cal A}}}^{(1)i\,{\rm
rest}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\omega,{\bf
n})=O(\omega^{-1+\epsilon})$ (I.10)

as $\omega\to 0$ with $\epsilon>0$. We will see that the second and third
terms in Eq. (I.8) give a vanishing contribution to the second order
scattering while the first term gives a nonvanishing contribution.

We next write the equation of motion (3.18a) for the second order scalar field
as $\Box{\underline{\Phi}}^{(2)}=s$, where the source $s$ is
$s=2i{\underline{A}}^{(1)\,a}\nabla_{a}{\underline{\Phi}}^{(1)}$. Defining the
final data on $\mathscr{I}^{+}$ as
${\underline{\chi}}^{(2)}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(u,{\bf
n})=\lim_{r\to\infty}r{\underline{\Phi}}^{(2)}(u+r,r{\bf n})$, we make use of
the general identity derived in Sec. 2.2 of Ref. [70]

${\tilde{\underline{\chi}}}^{(2)}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\omega,{\bf
n})=-2\pi^{2}{\tilde{s}}(\omega,\omega{\bf n}),$ (I.11)

where ${\tilde{s}}$ is the spacetime Fourier transform

${\tilde{s}}(\omega,{\bf k})=\frac{1}{(2\pi)^{4}}\int dt\int d^{3}xe^{i\omega
t}e^{-i{\bf k}\cdot{\bf x}}s(t,{\bf x})$ (I.12)

and the Fourier transform
${\tilde{\underline{\chi}}}^{(2)}_{\scalebox{0.7}{$\scriptscriptstyle+$}}$ is
defined as in Eq. (I.3) but with $v$ replaced by $u$. We now evaluate the
source $s(t,{\bf x})$ using the the mode expansions (I.1) and (I.4), and
multiply by a regulator factor $\exp[-t^{2}/(2T^{2})]$, where we will
eventually take the limit $T\to\infty$. We then evaluate the spacetime Fourier
transform, make use of the the identity (I.11), and eliminate the mode
functions in favor of the initial data on $\mathscr{I}^{-}$ using Eqs. (I.2)
and (I.6). This gives

$\displaystyle{\tilde{\underline{\chi}}}^{(2)}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\omega,{\bf
n})$ $\displaystyle=$
$\displaystyle-\lim_{T\to\infty}\sum_{\eta,\gamma=\pm}\int
d^{3}p\,\frac{\eta\gamma}{lp}\delta_{T}(\nu k-\eta l-\gamma p)$ (I.13)
$\displaystyle\times{\tilde{\underline{\chi}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}\left[\eta
l,-\eta\frac{({\bf k}-{\bf p})}{l}\right]\,{\bf
k}\cdot{\tilde{\bf{\underline{\cal
A}}}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}^{(1)}(\gamma p,-\gamma{\hat{\bf
p}}),$

where we have defined

${\bf k}=\omega{\bf n},\ \ \ k=|\omega|,\ \ \ \nu={\rm sgn}(\omega),\ \ \
l=|{\bf k}-{\bf p}|,$ (I.14)

and $\delta_{T}$ is the approximate $\delta$-function

$\delta_{T}(\omega)=\frac{T}{\sqrt{2\pi}}\exp\left[-\frac{1}{2}\omega^{2}T^{2}\right].$
(I.15)

To evaluate the momentum space integral (I.13) it is convenient to switch to
prolate spheroidal coordinates. We pick Cartesian coordinates for which ${\bf
k}=(0,0,k)$, and define coordinates $(\sigma,\tau,\varphi)$ with
$1\leq\sigma<\infty$, $-1\leq\tau\leq 1$, $0\leq\varphi\leq 2\pi$ by

${\bf
p}=\frac{1}{2}k(\sqrt{\sigma^{2}-1}\sqrt{1-\tau^{2}}\cos\varphi,\sqrt{\sigma^{2}-1}\sqrt{1-\tau^{2}}\sin\varphi,1+\sigma\tau).$
(I.16)

It follows that $d^{3}p=(k^{3}/8)(\sigma^{2}-\tau^{2})d\sigma d\tau d\varphi$
and that

$l=|{\bf k}-{\bf p}|=\frac{1}{2}k(\sigma-\tau),\ \ \ \
p=\frac{1}{2}k(\sigma+\tau).$ (I.17)

This gives

$\displaystyle{\tilde{\underline{\chi}}}^{(2)}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\omega,{\bf
n})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\lim_{T\to\infty}\sum_{\eta,\gamma=\pm}\int_{1}^{\infty}d\sigma\int_{-1}^{1}d\tau\int_{0}^{2\pi}d\varphi\,\eta\gamma\,\delta_{kT}\left[1-\frac{\eta(\sigma-\tau)}{2}-\frac{\gamma(\sigma+\tau)}{2}\right]$
(I.18)
$\displaystyle\times{\tilde{\underline{\chi}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}\left[\nu\eta
l,-\nu\eta\frac{{\bf k}-{\bf p}}{l}\right]\,{\bf
k}\cdot{\tilde{\bf{\underline{\cal
A}}}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}^{(1)}(\nu\gamma
p,-\nu\gamma{\hat{\bf p}}),$

where we have redefined $\eta\to\nu\eta$ and $\gamma\to\nu\gamma$.

We now discuss in turn the four different terms in the expression (I.18). Two
of these give vanishing contributions, and the contributions from the
remaining two are equal. For the first term with $\eta=\gamma=-1$, the
approximate delta function reduces in the limit $T\to\infty$ to
$\delta(\sigma+1)$, which gives a vanishing contribution since the range of
$\sigma$ is $\sigma\geq 1$. For the second term with $\eta=-1$, $\gamma=1$,
the approximate delta function reduces instead to $\delta(\tau-1)$, and the
expression reduces to an integral over the half line ${\bf
p}=(0,0,k+(\sigma-1)k/2)$ for $1\leq\sigma<\infty$. It follows that the
frequency argument $\nu\gamma p$ of ${\tilde{\bf{\underline{\cal
A}}}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}^{(1)}$ is bounded away from
zero, and therefore the second line of the integrand is bounded, from Eq.
(I.8); the soft degrees of freedom or distributional components do not
contribute. The Lorenz gauge condition (I.5) now makes the result vanish; this
is the essentially the Feynman diagram argument given in the last paragraph of
Sec. 5.

The third term with $\eta=\gamma=1$ is a little more involved. Inserting the
decomposition (I.8) of the initial data for the vector potential, this term
can be written as

$\displaystyle-\frac{1}{2}\lim_{T\to\infty}\int_{1}^{\infty}d\sigma\int_{-1}^{1}d\tau\int_{0}^{2\pi}d\varphi\,\delta_{kT}(1-\sigma)\,{\tilde{\underline{\chi}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}\left[\nu
l,-\nu\frac{{\bf k}-{\bf p}}{l}\right]$ $\displaystyle\times{\bf
k}\cdot\left[\frac{i\nu}{2\pi}{\bf\Delta{\underline{\cal
A}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(-\nu{\hat{\bf p}})\,{\rm
P.V.}\frac{1}{p}+{\bf{\overline{\underline{\cal
A}}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(-\nu{\hat{\bf
p}})\delta(p)+{\tilde{\underline{\cal A}}}^{(1)\,{\rm
rest}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\nu p,-\nu{\hat{\bf
p}})\right].$ (I.19)

Now the limiting $T\to\infty$ delta function $\delta(\sigma-1)$ would enforce
an integral over the line segment

${\bf p}=(0,0,k(1+\tau)/2)$ (I.20)

for $-1\leq\tau\leq 1$ that extends from ${\bf p}={\bf 0}$ to ${\bf p}={\bf
k}$ and that overlaps with the location ${\bf p}={\bf 0}$ of the
distributional contributions to the integrand. Thus we will need to take the
$T\to\infty$ limit carefully. For the second and third terms in the large
square brackets on the second line we can just use the limiting delta function
$\delta(\sigma-1)$. The third term scales as $\sim
p^{-(1-\epsilon)}\sim(1+\tau)^{-(1-\epsilon)}$ with $\epsilon>0$, from Eq.
(I.10). This yields a diverging integrand but the integral over $\tau$
converges, and thus the result vanishes by the Lorenz gauge condition (I.5),
since it is evaluated at ${\hat{\bf p}}={\hat{\bf k}}$ by Eq. (I.20).
Similarly, for the second term, the $\delta(\sigma-1)$ factor enforces
evaluation at ${\hat{\bf p}}={\hat{\bf k}}$, and then the $\delta(p)$ factor
evaluates at $\tau=-1$ at which the integrand is finite, giving again a
vanishing result by Eq. (I.5).

Thus we are left with the first term in the large square brackets in the
expression (I.19). We simplify this in several stages:

  * •

First, we argue that we can evaluate the factor
${\tilde{\underline{\chi}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ at
${\bf p}={\bf 0}$ and pull it outside the integral. To see this we define the
variable

$\mu={\hat{\bf k}}\cdot{\hat{\bf p}}=\frac{1+\sigma\tau}{\sigma+\tau}$ (I.21)

from Eq. (I.16), and we note that by the Lorenz gauge condition (I.5) that
${\bf k}\cdot\Delta{\underline{\cal
A}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(-\nu{\hat{\bf p}})$ must
vanish at $\mu=1$ and at $\mu=-1$. Hence it must be expressible as $1-\mu^{2}$
times a bounded function, and including the factor of $1/p$ and using Eqs.
(I.17) and (I.21) we find that the second line in the expression (I.19) can be
written as a bounded function times the factor

$\frac{(\sigma^{2}-1)(1-\tau^{2})}{(\sigma+\tau)^{3}}.$ (I.22)

Now the approximate delta function $\delta_{kT}(1-\sigma)$ will restrict the
value of $\sigma$ to $\sigma-1=O(\varepsilon_{0})$ where
$\varepsilon_{0}=1/(kT)$ and we are taking the limit $\varepsilon_{0}\to 0$,
from Eq. (I.15). Thus when $\tau+1=O(1)$ the factor (I.22) scales like
$\sim\varepsilon_{0}$ and the integrand is suppressed, whereas when
$\tau+1=O(\varepsilon_{0})$ the factor (I.22) scales like
$\sim\varepsilon_{0}^{-1}$. Hence for small $\varepsilon_{0}$ we can evaluate
the factor
${\tilde{\underline{\chi}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ in
the expression (I.19) at $\tau=-1$ and pull it outside the integral, where it
can be written as
${\tilde{\underline{\chi}}}_{\scalebox{0.7}{$\scriptscriptstyle-$}}^{(1)}(\omega,-{\bf
n})$ by Eq. (I.14).

  * •

Second, we can drop the integral over $\sigma$ and the approximate delta
function, evaluate the remaining integral at $\sigma=1+\varepsilon_{0}$, take
the limit $\varepsilon_{0}\to 0$, and then multiply by a correction factor of
$1/2$. This is because we can pull the approximate delta function outside the
integral over $\tau$ and $\varphi$, and that integral becomes independent of
$\sigma$ in the limit $\sigma\to 1$ from above (see below). Then for small
$\varepsilon_{0}$ the integral over $\sigma$ of the approximate delta function
gives an overall factor of $1/2$, since the range of $\sigma$ is restricted to
$\sigma\geq 1$.

  * •

Third, the remaining integral at a fixed value of $\sigma$ is an integral over
a prolate spheroid that limits as $\sigma\to 1$ to the line segment (I.20). We
parameterize this integral as an integral over solid angles centered at ${\bf
p}={\bf 0}$ by changing one of the variables of integration from $\tau$ to
$\mu$, using

$\frac{1}{\tau+\sigma}d\tau=\frac{1}{\sigma-\mu}d\mu$

from Eq. (I.21).

The result of all these simplifications is the expression

$-\frac{i}{4\pi}{\tilde{\underline{\chi}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\omega,-{\bf
n})\,\lim_{\sigma\to
1}\,\int_{-1}^{1}d\mu\int_{0}^{2\pi}d\varphi\,\frac{1}{\sigma-\mu}\,\nu{\hat{\bf
k}}\cdot\Delta{\underline{\cal
A}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(-\nu{\hat{\bf p}})$ (I.23)

for the $\eta=\gamma=1$ piece of Eq. (I.18). Defining the unit vectors
${\hat{\bf m}}=-\nu{\hat{\bf p}}$ and ${\hat{\bf m}}_{0}=-\nu{\hat{\bf
k}}=-{\bf n}$ from the definitions (I.14), and taking the limit $\sigma\to 1$,
we can rewrite this as

$\frac{i}{4\pi}{\tilde{\underline{\chi}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\omega,-{\bf
n})\,\,\int d^{2}\Omega_{\hat{\bf m}}\,\frac{{\hat{\bf
m}}_{0}\cdot\Delta{\underline{\cal
A}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}({\hat{\bf m}})}{1-\mu}.$
(I.24)

Next we define the Green’s function

$G({\hat{\bf m}},{\hat{\bf m}}_{0})=\frac{1}{4\pi}\log(1-{\hat{\bf
m}}\cdot{\hat{\bf m}}_{0})$ (I.25)

which satisfies $D^{A}D_{A}G=\delta^{2}({\hat{\bf m}},{\hat{\bf
m}}_{0})-1/(4\pi)$. Using Eq. (I.9) we can rewrite the expression (I.24) as

$-i{\tilde{\underline{\chi}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\omega,-{\bf
n})\,\int d^{2}\Omega_{\hat{\bf m}}\,D^{A}G({\hat{\bf m}},{\hat{\bf
m}}_{0})D_{A}\Delta{\underline{\Psi}}^{{\rm
e}(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}({\hat{\bf m}}),$ (I.26)

where we have used $D_{A}m^{i}=e^{i}_{A}$. Finally integrating by parts and
using the fact that $\Delta{\underline{\Psi}}^{{\rm
e}(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}$ has no $l=0$ component [cf.
Eq. (2.36b) above] gives that the contribution to
${\tilde{\underline{\chi}}}^{(2)}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\omega,{\bf
n})$ is

$i{\tilde{\underline{\chi}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\omega,-{\bf
n})\Delta{\underline{\Psi}}^{{\rm
e}(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(-{\bf n}).$ (I.27)

We now turn to the fourth and final term $\eta=1$, $\gamma=-1$ in Eq. (I.18).
This can be evaluated using the same methods as for the $\gamma=\eta=1$ term
just described. One ends up integrating over a hyperboloid of revolution of
the form $\tau=-1+O(\varepsilon_{0})$ which limits to the half line ${\bf
p}=(0,0,(1-\sigma)k/2)$ for $1\leq\sigma<\infty$. The final result is the same
as the result (I.27), and adding the two contributions gives

${\tilde{\underline{\chi}}}^{(2)}_{\scalebox{0.7}{$\scriptscriptstyle+$}}(\omega,{\bf
n})=2i{\tilde{\underline{\chi}}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(\omega,-{\bf
n})\Delta{\underline{\Psi}}^{{\rm
e}(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(-{\bf n}),$ (I.28)

which is equivalent to Eq. (5.4).

The remaining relations in Eqs. (5.3) can be evaluated using similar methods;
all the second order contributions vanish since they are quadratic in the
incoming first order scalar field by Eq. (3.18b), and that scalar field has no
soft piece.

##  Appendix J Leading order change in electromagnetic memory

In this appendix we compute explicitly the leading order electromagnetic
memory produced when there is an incoming scalar field but no incoming vector
potential. From the conservation law (3.36) this quantity is directly related
to the change $\Delta Q^{\rm hard}({\bf n})$ between the total hard outgoing
charge per solid angle at $\mathscr{I}^{+}$ and that at $\mathscr{I}^{-}$.
This quantity is the lowest order coupling of soft and hard degrees of freedom
that cannot be removed by modifying the definitions of the hard and soft
sectors, as demonstrated in Sec. 6. We shall show that it is non-zero in
general.

The quantity $\Delta Q^{\rm hard}({\bf n})$ is quartic in the incoming Lorenz
gauge field
${\underline{\chi}}^{(1)}_{\scalebox{0.7}{$\scriptscriptstyle-$}}(v,{\bf n})$.
For any function $\varepsilon=\varepsilon({\bf n})$ we define $\Delta Q^{\rm