$J_{mn} := L_{mn} + S_{mn}$ is given by

\begin{align}
\label{eq:def-JAM}
% J_{\da\db} = z^{\dc\c} p_{\smash{\c(\db}\vphantom{\da)}} \be_{\da)\dc}
% \,,\quad
% J^{\a\b} = p_{\c\dc} \bz^{\dc(\b} \e^{\a)\c}
% \,.
= - z^{\da\a} p_{\a\db} + \minie\mem \delta^\da{}_\db\hem (z^{\dc\c}p_{\c\dc})
\,,\quad
\bar{J}_\a{}^\b
= - p_{\a\da} \bz^{\da\b} + \minie\mem \delta_\a{}^\b\hem \hnem(p_{\c\dc}\bz^{\dc\c})
\,.
\end{align}

Since $L_{mn}$, $S_{mn}$, and $J_{mn}$ are all conserved quantities in the free theory, we can use the interaction-picture expressions for computing their impulse.

First, we find from (<ref>) that

\begin{align}
\label{eq:exactimp.S}
% \begin{split}
\renewcommand{\arraystretch}{1.1}
\begin{array}{rl}
\mathit{\Delta}\bar{S}_\a{}^\b = 0
\,,\quad
\mathit{\Delta}S^\da{}_\db
&{}= \M^+\,
\deltabar(2\mem\three{\mem\cdot\mem}\one)\,
% \big({
%     (a^{\da\a}\one_{\a\dc}) (\bthree^\dc \bthree_\db/mx)
%     - (\bthree^\da \bthree_\dc/mx) (a^{\dc\c}\one_{\c\db})
% }\big)
\big(
|\bthree\rsq\lsq\bthree| a|\one|
- |a|\one |\bthree\rsq\lsq\bthree|
\big){}^\da{}_\db/mx
\,,\\
&{}= \M^+\,
\deltabar(2\mem\three{\mem\cdot\mem}\one)\,
\three^{\da\a}a_{\smash{\a\db}\vphantom{\b}}
- a^{\da\a}\three_{\smash{\a\db}\vphantom{\b}}
\,,\\
&{}= i\hem \M^+\,
\deltabar(2\mem\three{\mem\cdot\mem}\one)\,
(-\ve_{mnrs} \hem a^r\hnem \three^s)\hem
\,,
\end{array}}
% \end{split}
\end{align}

which is consistent with the chiral Lorentz transformation interpretation.

Next, we compute $\mathit{\Delta}J^\da{}_\db$.
$(\t_* {-\mem} \t_0)\mem L^\da{}_\db(\t,\t_0)\hem u^{\db\a}
\propto (p^{-1}\hnem(\t))^{\da\a}$
term in (<ref>)
does not boil down to a particularly neat expression in the infinite time
limit, but it

drops out when
one computes
$p_{\a\da}\hnem(\t)\mem \smash{\underset{\adjustbox{raise=0.2em,scale=0.8}{$*$}}{z}{}^{\da\b}(\t)}$
$\smash{\underset{\adjustbox{raise=0.2em,scale=0.8}{$*$}}{z}{}^{\da\a}(\t)}\mem p_{\smash{\a\db}\vphantom{\a\da}}\hnem(\t)$
and symmetrizes the
two spinor
As a result,
we find

\begin{align}
\begin{split}
\label{eq:exactimp.am}
\mathit{\Delta}J^{\smash{\da\db}\vphantom{(\da}}
\mem&=\mem
i\hem \M^+\mem \deltabar(2\mem\three{\mem\cdot\mem}\one)\mem
% z_{\smash{*}}^{(\da\a}
% \three_\a \bthree\vphantom{z}^{\smash{\db)}\vphantom{(\da}}
\big({
- z_{\smash{*}}^{(\da\a}
\hem \three_\a\vphantom{z}^{\smash{\db)}\vphantom{(\da}}
\,,\\
\mathit{\Delta}L^{\smash{\da\db}\vphantom{(\da}}
\mem&=\mem
i\hem \M^+\mem \deltabar(2\mem\three{\mem\cdot\mem}\one)\mem
% (z_{\smash{*}} \hem{+}\mem 2ia)^{(\da\a}
% \three_\a \bthree\vphantom{z}^{\smash{\db)}\vphantom{(\da}}
\big({
- (z_{\smash{*}} \hem{+}\mem 2ia)^{(\da\a}
\hem \three_\a\vphantom{z}^{\smash{\db)}\vphantom{(\da}}
\,,\\
\mathit{\Delta}\bar{J}^{\smash{\a\b}\vphantom{(\da}}
\mem=\mem
\mathit{\Delta}\bar{L}^{\smash{\a\b}\vphantom{(\da}}
\mem&=\mem
i\hem \M^+\mem \deltabar(2\mem\three{\mem\cdot\mem}\one)\,
\big({
- \three^{(\a}{}_\da
\hem z_{\smash{*}}^{\da\b)}
\,,
% \,.
\end{split}
\end{align}

noting that

\begin{align}
% G^\da{}_\db(\t,\t_0)
% =
% \frac{\nu}{2m}\mem \bthree^\da \bthree_\db\mem
%     \mathe^{i\hem \three\cdot z(\t_0)}
%     \nem\nem\left[\hem{
%         \frac{\partial}{\partial\alpha}\nem\hnem
%         \int_{\t_0}^{\t} \hspace{-0.2em}d\t'\,
%         \mathe^{i\alpha(\t'-\t_0)}
%     }\hem\right]_{\alpha = 3\cdot u}\nem
= (2/x)\mem \M^+\hem
\deltabar'\hnem(2\mem\three{\mem\cdot\mem}\one)\,
\bthree^\da \bthree_\db\mem
% \deltabar'\hnem(2\mem\three{\mem\cdot\mem}\one)
= (2/m)\mem \M^+\hem
\deltabar'\hnem(2\mem\three{\mem\cdot\mem}\one)\,
\one^{\da\a} \three_{\smash{\a\db}\vphantom{\da}}
% \mem\deltabar'\hnem(2\mem\three{\mem\cdot\mem}\one)
\,.
\end{align}

We have defined

\begin{align}
\label{eq:deflection-intersection}
\mem:=\mem (b-ia)^{\da\a}
+ u^{\da\a}\mem T
% + u^{\da\a}
% \nem\hnem
% \left(
%     \frac{-im}{\three{\mem\cdot\mem}\one}
%     - u \cdot z({-}\infty)
% \right)
% \,.
\,,\quad
% - u \cdot z({-}\infty)
% \frac{-im}{\three{\mem\cdot\mem}\one}
- u \cdot z({-}\infty)
% \,.
\,,
\end{align}

which has a natural interpretation as
the intersection point of the initial and final asymptotic holomorphic worldlines.
The existence of such a point is not a trivial fact
and is a consequence of the special geometry of this scattering problem.

To sum up,
(<ref>) boils down to

\begin{align}
\begin{split}
\label{eq:amimpulses}
% \label{eq:amimpulse.S}
\mathit{\Delta}S^{mn} &=
i\hem \M^+\mem \deltabar(2\mem\three{\mem\cdot\mem}\one)\mem
\big(\mem{
\hnem- 2i\mem a^{[m} \three^{n]}
- \ve^{mnrs} a_r \three_s
\,,\\
% \label{eq:amimpulse.L}
\mathit{\Delta}L^{mn} &=
i\hem \M^+\mem \deltabar(2\mem\three{\mem\cdot\mem}\one)\mem
\big(\mem{
2\mem \one^{[m} \three^{n]} (T/m)
+ 2b^{[m} \three^{n]}
+ \ve^{mnrs} a_r \three_s
\,,\\
% \label{eq:amimpulse.J}
\mathit{\Delta}J^{mn} &=
i\hem \M^+\mem \deltabar(2\mem\three{\mem\cdot\mem}\one)\mem
\big(\mem{
2\mem \one^{[m} \three^{n]} (T/m)
+ 2b^{[m} \three^{n]}
- 2i\mem a^{[m} \three^{n]}
\,.
\end{split}
\end{align}

Note that only $\mathit{\Delta}J^{mn} = i\hem\M^+\mem \deltabar(2\mem\three{\mem\cdot\mem}\one)\, z_{\smash{*}}^{[m} \three^{n]}$ is “polar.”
The other two have “axial” components,
i.e., terms involving the epsilon tensor.

Especially, the orbital angular momentum impulse $\mathit{\Delta}L^{mn}$ is not polar.
One may interpret this fact
as a failure of a local particle interpretation
in real spacetime
(and the necessity of complex spacetime).

Exact √Kerr impulses

Finally, let us re-solve the equation of motion with (<ref>).
We start off with .
For clarity's sake, we spell out the entire symplectic form:

\begin{align}
\label{eq:pws.sqrt-Kerr}
\begin{array}{rl}
\omega_{\nem{\Kerr}}^{\text{SDPW}}
\,=
i\mem d\bZ_I{}^\rmA \swedge dZ_\rmA{}^I
+ i\hem d\bpsi_i \swedge d\psi^i
\\
{}+ \sqrt{2}g\, dq_a c^a \swedge \lsq\bo|dz|\i\rangle
\mem \mathe^{-i\hem\bozo}
+ {\textstyle\frac{ig}{\sqrt{2}}}\mem q_a c^a \mem
\bo_\da\hem\bo_\db (dz \swedge dz)^{\da\db}
\mem \mathe^{-i\hem\bozo}
\,.
\end{array}}
\end{align}

The resulting equation of motion boils down to

\begin{align}
\begin{split}
\label{eq:xeom+1a}
\dot{\lambda}_\a{}^I
= {\textstyle\frac{1}{2i}} (\lambda \hem \s_a)_\a{}^I
\dot{\rambda}_{I\da}
= (i\nu{\hnem/}2m)\mem
\rambda_{\smash{I\db}\vphantom{I}} \mem \bo^\db \bo_\da
\mem\mathe^{-i\hem\bozo}
- {\textstyle\frac{1}{2i}} (\s_a \rambda)_{I\da}
\,,\\
\dot{z}^{\da\a} = \minime m(p^{-1})^{\da\a}
\dot{y}^{\da\a}
= -(i\nu{\hnem/}2m)\mem
\bo^\da \bo_\db \mem y^{\db\a}
\mem\mathe^{-i\hem\bozo}
\,,
\end{split}\\
\label{eq:xeom+1b}
\dot{\psi}^i
= -({\textstyle\frac{g}{\sqrt{2}}}\tilde{x}\hem c^a)\mem
(t_a)^i{}_j \hhem \psi^j \hem
% \mem(\langle\i|p|\bo\rsq/m)
\mem\mathe^{-i\hem\bozo}
\dot{\bar{\psi}}_i
= ({\textstyle\frac{g}{\sqrt{2}}}\tilde{x}\hem c^a)\mem
\bpsi_j \hhnem (t_a)^j{}_i \hem
% \mem(\langle\i|p|\bo\rsq/m)
\mem\mathe^{-i\hem\bozo}
% \,,
\,.
\end{align}

The last line implies
= (q_b f^b{}_{ca}\hhem c^c)\mem
\,\mathe^{-i\hem\bozo}$.
We have defined the “off-shell $x$-factor”

\begin{align}
\tilde{x} := -\langle\i|p|\bo\rsq/|\Delta|
\,,
\end{align}

which is a constant of motion.
Note that both “holomorphic” and “anti-holomorphic” color degrees of freedom,
$\psi^i$ and $\bpsi_i$,
are localized at the holomorphic position $z^{\da\a}$,
being parallel-transported by the holomorphic gauge connection.

both $\psi^i$ and $\bpsi_i$
should couple to the holomorphic connection
for $q_a F^a\hnem(z) = (i\hem \bpsi \hem t_a \psi) F^a\hnem(z)$ to be gauge invariant.

We can understand (<ref>) as a combination of
an infinitesimal self-dual null rotation, color rotation, and internal $\mathrm{SU}(2)$ rotation
with fixed points $\bo^\da$, $c^a$, and $\Omega^a$,
We believe that it is clear from the context whether
the indices $a,b,c,\cdots$ denote the color indices or the little group indices.
Hence, $\nu = \sqrt{2}g\mem q_ac^a$ is conserved.

Now, since (<ref>) is totally unaffected by the $dq \swedge dz$ correction,
it follows that
the solution (<ref>)
and the impulses (<ref>)-(<ref>)
are all valid.
The only additional piece of information
is the color time evolution and impulse
that follow from (<ref>):

\begin{align}
\psi^i\hnem(\t)
&= U^i{}_j(\t,\t_0)\mem \psi^j\hnem(\t_0)
\,,\quad
\bpsi_i(\t)
= \bpsi_j(\t_0)\mem U^j{}_i(\t_0,\t)
\,,\\
% \stepcounter{parentequation} \gdef\theparentequation{\arabic{section}.\arabic{parentequation}} \setcounter{equation}{0}
% \mathit{\Delta}\psi^i
% =
% &\,,\quad
\label{eq:pwH-colorphase}
\exp\nem\nem\left(\hem{
\hnem\frac{g\tilde{x}}{\sqrt{2}}\mem c^a(t_a)\,
\mathe^{i\three\cdot(z(\t_0)-u\t_0)}
\nem\nem\int_{\t_1}^{\t_2}
\hspace{-0.2em} d\t'\,\mem
\mathe^{
i (\three\cdot u)
\t'
\nem\nem{\vphantom{\Big|}}^i{\vphantom{\Big|}}_j
\,,\\
\label{eq:inftylimit-U}
\exp\nem\nem\left(\mem{
i\hem \M^{+1}\nem(\one,a;\three)\mem
\mathe^{i\three\cdot b}\mem
\deltabar(2\mem\three{\mem\cdot\mem}\one)
\mem c^a\mem i(t_a)
\nem\hnem{\vphantom{\big|}}^i{\vphantom{\big|}}_j
\,.
\end{align}

Note that $\smash{\underset{\adjustbox{raise=0.2em,scale=0.8}{$*$}}{
\psi
}{\vphantom{\psi}}^i\hnem(\t)} = \psi^i\hnem(\t)$
\bpsi
}{\vphantom{\bpsi}}_i(\t)} = \bpsi_i(\t)$,
since $\dot{\psi}^i = 0$ and $\dot{\bar{\psi}}_i = 0$ in the free theory.

The color rotation (<ref>) may be thought of as the “square root” of
the chiral Lorentz transformation (<ref>)
or the

additional null translation
which we derive shortly.

The time evolution and the impulse of the color charge $q_a$
easily follow from translating (<ref>)-(<ref>)
to the adjoint representation.
One may expand the exponential and find

\begin{align}
\begin{split}
\label{eq:colorimpulses}
\mathit{\Delta}\psi^i
\mem&=\mem i\hem \M^{+1}\nem(\one,a;\three)\mem
\mathe^{i\three\cdot b}\mem
\deltabar(2\mem\three{\mem\cdot\mem}\one)\,
(\hhem (t_c)^i{}_j\hem \psi^j \hhem)\mem ic^c
+ \mathcal{O}(g^2)
\,,\\
\mathit{\Delta}\bpsi_i
\mem&=\mem i\hem \M^{+1}\nem(\one,a;\three)\mem
\mathe^{i\three\cdot b}\mem
\deltabar(2\mem\three{\mem\cdot\mem}\one)\,
(\hhem -\bpsi_j (t_c)^j{}_i \hhem)\mem ic^c
+ \mathcal{O}(g^2)
\,,\\
\mathit{\Delta}q_a
\mem&=\mem i\hem \M^{+1}\nem(\one,a;\three)\mem
\mathe^{i\three\cdot b}\mem
\deltabar(2\mem\three{\mem\cdot\mem}\one)\,
(\hhem -q_b\hem f^b{}_{ca} \hhem)\mem ic^c
+ \mathcal{O}(g^2)
\,.
\end{split}
\end{align}

Exact Kerr impulses

For Kerr in the self-dual gravitational plane wave,

the symplectic form is
given by

\begin{align}
\begin{split}
\label{eq:pws.Kerr}
\renewcommand{\arraycolsep}{0em}
\begin{array}{rl}
\omega_\text{Kerr}^\text{SDPW}
\,=\,\mem\hhem
i\mem d\bZ_I{}^\rmA \swedge dZ_\rmA{}^I
\\
- \frac{1}{\MPl}\mem
\langle\i|dp|\bo\rsq {\hem\wedge\hem} \lsq\bo|dz|\i\rangle
\mem\mathe^{-i\hem\bozo}
+ \frac{im\tilde{x}}{2\MPl}\mem
\bo_\da\hem\bo_\db\hhem(dz \swedge dz)^{\da\db}
\mem\mathe^{-i\hem\bozo}
\,.
\end{array}}
\end{split}
\end{align}

The equation of motion reads

\begin{align}
\begin{split}
\label{eq:xeom+2}
\dot{\rambda}_{I\da}
= (i\nu{\hnem/}2m)\mem
\rambda_{\smash{I\db}\vphantom{I}} \mem \bo^\db \bo_\da
\mem\mathe^{-i\hem\bozo}
- {\textstyle\frac{1}{2i}} (\s_a \rambda)_{I\da}
\dot{\lambda}_\a{}^I
= {\textstyle\frac{1}{2i}} (\lambda \hem \s_a)_\a{}^I
\,,\\
% \dot{z}^{\da\a}
% = \minime m(p^{-1})^{\da\a}
% + (\tilde{x}\mem{\bo^\da\i^\a}\nem\nem/2\MPl)
% \mem\mathe^{-i\hem\bozo}
\dot{z}^{\da\a}
= \minime\big(\hem{
{\hem-\mem} (\nu{\hnem/\hhnem}m)\mem{\bo^\da\i^\a}
\mem\mathe^{-i\hem\bozo}
\dot{y}^{\da\a}
= -(i\nu{\hnem/}2m)\mem
\bo^\da \bo_\db \mem y^{\db\a}
\mem\mathe^{-i\hem\bozo}
\,.
\end{split}
\end{align}

The additional “zitterbewegung” of $z^{\da\a}$ lies within the $\da$-plane of $|\bo\rsq$.
As a result,
$\nu {\mem=\mem}$ $(m{\hhnem/\hnem}\MPl)\hem\tilde{x}$ is conserved, and
(<ref>) is valid.
This means that the solution (<ref>) holds except for $z^{\da\a}\hnem(\t)$.
The revised solution for $z^{\da\a}\hnem(\t)$ is given as

\begin{align}
\label{eq:zsol-fluc}
\mem=\mem z^{\da\a}\hnem(\t_0)
+ F^\da{}_\db(\t,\t_0)\mem u^{\db\a}
+ X^{\da\a}\hnem(\t,\t_0)
\,,
\end{align}

where the new contribution $X^{\da\a}\hnem(\t,\t_0)$ is given by

\begin{align}
\label{eq:pwH-X}
\mem:={}&\mem
\frac{\tilde{x} \mem\bo^\da\i^\a}{2\MPl}\mem
\mathe^{i\three\cdot(z(\t_0)-u\t_0)}
\nem\nem\int_{\t_0}^{\t}
\hspace{-0.2em} d\t'\,\mem
\mathe^{
i (\three\cdot u)
\t'
\,,\\
\label{eq:Delta-X}
\iq
\mem={}&\mem
\M^{+2}\hnem(\one,a;\three)\mem
\mathe^{i\three\cdot b}\mem
\deltabar(2\mem\three{\mem\cdot\mem}\one)\,
\,.
\end{align}

Accordingly, we have

\begin{align}
\label{eq:z*-fluc}
% \iq
\smash{\underset{\adjustbox{raise=0.2em,scale=0.8}{$*$}}{z}{}^{\da\a}\hnem(\t)}
\mem&=\mem
+ ((\t_* {-\mem} \t_0)\hem L^\da{}_\db(\t,\t_0)
- G^\da{}_\db(\t,\t_0))\mem u^{\db\a}
+ X^{\da\a}\hnem(\t,\t_0)
\,.
\end{align}

Finally, it follows
from (<ref>) and (<ref>) that

\begin{align}
\label{eq:timedifference-fluc}
\t_*(\t_0,\t)
= \t_0
+ \frac{1}{m}\hem
z^{\da\a}\hnem(\t_0)\mem p_{\a\db}(\t_0)\mem L^\db{}_\da(\t_0,\t)
- \frac{\tilde{x}^2}{2\MPl}\mem
\mathe^{i\three\cdot(z(\t_0)-u\t_0)}
\nem\nem\int_{\t_0}^{\t}
\hspace{-0.2em} d\t'\,\mem
\mathe^{
i (\three\cdot u)
\t'
\,,
\end{align}

from which we obtain

\begin{align}
\label{eq:timed*-fluc}
\delta\t_*
\mem&=\mem
\M^{+2}\hnem(\one,a;\three)\mem \mathe^{i\three\cdot b}\mem
\deltabar(2\mem\three{\mem\cdot\mem}\one)\,
% \mem\hhem
% \,,\\
% \label{eq:z*-fluc}
% % \iq
% \smash{\underset{\adjustbox{raise=0.2em,scale=0.8}{$*$}}{z}{}^{\da\a}\hnem(\t)}
% \mem&=\mem
% z^{\da\a}\hnem(\t_0)
% + ((\t_* {-\mem} \t_0)\hem L^\da{}_\db(\t,\t_0)
% - G^\da{}_\db(\t,\t_0))\mem u^{\db\a}
% + X^{\da\a}\hnem(\t,\t_0)
\,.
\end{align}

The additional term $X^{\da\a}\hnem(\t,\t_0)$ does not affect
$\mathit{\Delta}p_{\a\da}$ and $\mathit{\Delta}y^{\da\a}$,
but changes occur in $\mathit{\Delta}\lambda_\a{}^I$, $\mathit{\Delta}\rambda_{I\da}$, and $\mathit{\Delta}z^{\da\a}$.

First, the expression (<ref>)
for the spin frames
holds with (<ref>) instead of (<ref>).
Second, the angular momentum impulses (<ref>) change to

\begin{align}
\label{eq:exactimp.am-X}
\mathit{\Delta}J^{\smash{\da\db}\vphantom{(\da}}
\mem=\mem {}&{}
i\hem \M^{+2}\hnem(\one,a;\three)\mem \mathe^{i\three\cdot b}\mem \hhem\deltabar(2\mem\three{\mem\cdot\mem}\one)\mem
\big({
- z_{\smash{*}}^{(\da\a}
\hem \three_\a\vphantom{z}^{\smash{\db)}\vphantom{(\da}}
+ i\mem \bo^{\smash{(\da}\vphantom{(\da}}\bi^{\smash{\db)}\vphantom{(\da}}
+ K^{\smash{\da\db}\vphantom{(\da}}\hhem
% - X^{\da\a}\hnem({+}\infty,{-}\infty)\mem p_{\smash{\a\db}\vphantom{\da}}({+}\infty)
\,,\\
\nonumber
\mathit{\Delta}L^{\smash{\da\db}\vphantom{(\da}}
\mem=\mem {}&{}
i\hem \M^{+2}\hnem(\one,a;\three)\mem \mathe^{i\three\cdot b}\mem \hhem\deltabar(2\mem\three{\mem\cdot\mem}\one)\mem
\big({
- (z_{\smash{*}} \hem{+}\mem 2ia)^{(\da\a}
\hem \three_\a\vphantom{z}^{\smash{\db)}\vphantom{(\da}}
+ i\mem \bo^{\smash{(\da}\vphantom{(\da}}\bi^{\smash{\db)}\vphantom{(\da}}
+ K^{\smash{\da\db}\vphantom{(\da}}\hhem
% - X^{\da\a}\hnem({+}\infty,{-}\infty)\mem p_{\smash{\a\db}\vphantom{\da}}({+}\infty)
\,,\\
\nonumber
\mathit{\Delta}\bar{J}^{\smash{\a\b}\vphantom{(\da}}
\mem=\mem
\mathit{\Delta}\bar{L}^{\smash{\a\b}\vphantom{(\da}}
\mem=\mem {}&{}
i\hem \M^{+2}\hnem(\one,a;\three)\mem \mathe^{i\three\cdot b}\mem \hhem\deltabar(2\mem\three{\mem\cdot\mem}\one)\,
\big({
- \three^{(\a}{}_\da
\hem z_{\smash{*}}^{\da\b)}
% + \bar{K}^{\smash{\a\b}\vphantom{(\da}}\hhem
-i\mem \o^{\smash{(\a}\vphantom{(\da}}\i^{\smash{\b)}\vphantom{(\da}}
% - p_{\smash{\a\da}\vphantom{\da}}({+}\infty)\mem X^{\da\b}\hnem({+}\infty,{-}\infty)
\,,\\
\label{eq:Kam}
\mem:=\mem {}&{}
% i\mem \bo^{\smash{(\da}\vphantom{(\da}}\bi^{\smash{\db)}\vphantom{(\da}}
% +
\big(\hem{
\M^{+2}\hnem(\one,a;\three)\mem \mathe^{i\three\cdot b}\mem \hhem\deltabar(2\mem\three{\mem\cdot\mem}\one)
+i\mem \langle\i|\one|\bi\rsq
% \,,\\
% \bar{K}^{\smash{\a\b}\vphantom{(\da}}
% &=
% -i\mem \o^{\smash{(\a}\vphantom{(\da}}\i^{\smash{\b)}\vphantom{(\da}}
\,.
\end{align}

Since the $\t_*$ term drops out in the angular momenta,
the definition (<ref>) of $z_{\smash{*}}^{\da\a}$ still applies to (<ref>).

However, the intersection point
no longer exists
because $K^{\smash{\da\db}\vphantom{(\da}}$ makes $\mathit{\Delta}J^{mn}$ no longer polar.

Overall, the heavenly plane-wave scattering problem of /Kerr
is a maximally extended system
of interlocking commuting transformations:
self-dual null rotation,
holomorphic self-dual null translation,
color and little group rotations
with a fixed point.

§ SUMMARY AND OUTLOOK

Let us recapitulate the main ideas and results we have presented.
The inputs of this paper were
symplectic perturbation theory [82],
the massive twistor description of a free massive spinning particle in four dimensions [1].

* The “symplectic perturbation theory” of a particle [82]

to background fields
as perturbations on the

Poisson bracket
while retaining the same Hamiltonian.
From the geometric series expansion (<ref>),
one perturbatively

Poisson bracket and
the Hamiltonian equation of motion
in the orders of the

symplectic perturbation.
The amplitude
is then
the Born approximation.

Lorentz symmetry and little group symmetry
uniquely determine
the kinematics of a massive spinning particle:
the geometry of the free theory's physical phase space [1].

A hidden Kähler (“zig-zag”)
structure of the phase space
manifests in the massive twistor description

unifies spin and spacetime into “spin-space-time”
and redescribes
momentum and the body frame with massive spinor-helicity variables.

Applying symplectic perturbation theory
to the massive ambitwistor space,
we aimed to construct a formulation
of interacting massive spinning particles
that respects the remarkable Kähler property of the free theory.
It turned out that
the zig-zag structure of the free theory (“kinematics”)
tells us a lot about the interacting theory (“dynamics”).

* Via the zig-zag structure,
spin precession behavior
the spin-space-time part of the symplectic perturbation.

* (Newman-Janis shift derived)
the fact that the minimal spin precession equation of motion under a self-dual background
is given by $\dot{\lambda}_\a{}^I\nem = 0$
implies that self-dual field strengths continue holomorphically into the
complexified Minkowski space
if the particle is minimally coupled.

* (Geometrical interpretation of the Wilson coefficients)

Levi and Steinhoff [26]'s
spin multipole
Wilson coefficients

non-holomorphic continuations
of the self-dual “field strength”
into the spin-space-time.

* The zig-zag diagrammatics

manifests the minimal nature of the Kerr-Newman coupling
as well as the non-minimal nature of generic couplings.
The number of terms in the zig-zag representation of the perturbed

at $n$th order
given as the following.

generic couplings

minimal or
antipodal minimal

Earthly background

Heavenly background
expansion truncates

Not only gaining these insights,
we also have obtained new results:
a) a complete generalization of the TBMT and MPTD equations up to all orders in spin
and its matching with the three-point on-shell amplitude,
b) exact symplectic structures of the  and Kerr particles in self-dual backgrounds
and exact solutions to their equations of motion.

* A dictionary between
spin precession
equations of motion,
spin frame impulses,

three-point amplitudes
by taking the
symplectic perturbation as a common root.

* The non-spin-space-time part of the symplectic perturbation can be deduced from demanding

the closure of the symplectic form.

* Exact expressions of
and Kerr symplectic structures
in self-dual plane-wave backgrounds
are bootstrapped
from the matching (<ref>)
and the symplectivity requirement.
The resulting equations of motion
can be
analytically solved.
In turn,

amplitudes and impulses
are obtained as exact quantities.

* Further,
symplectic structures of
and Kerr
in generic self-dual backgrounds
are deduced from
the derivative counting (<ref>):

\begin{align}
\begin{split}
\label{eq:disc.Kerrs}
\text{{\Kerr}}:\quad
\omega
i\mem d\bZ_I{}^\rmA \swedge dZ_\rmA{}^I
+ i\mem d\bpsi_i \swedge d\psi^i
% + d(\hhem
%     q_a\hem A^a{}_m\hnem(z)\mem dz^m
% \hhem)
+ d\Big(\hem\mem{
A^a{}_m\hnem(z)\mem dz^m
\,,\\
\text{Kerr}:\quad
\omega
i\mem d\bZ_I{}^\rmA \swedge dZ_\rmA{}^I
+ d\Big(\mem{
-\rambda_{I\da} (\bs_m)^{\da\a} \lambda_\a{}^I\hem
h^m{}_n\hnem(z)\mem dz^n
\,.
% \,,
\end{split}
\end{align}

For the reader's convenience,
let us provide a summary of the “dictionary” here.
When the spin-space-time part of the self-dual symplectic perturbation is given as

\begin{align}
\label{eq:disc.omega-generic}
\omega'^+
(\minie\mem \wedgetwo{z}{z}{m}{n})
\wedgetwo{z}{\bz}{m}{n}
(\minie\mem \wedgetwo{\bz}{\bz}{m}{n})
% \,.
\,,
\end{align}

where each component is controlled by the Wilson coefficients as (<ref>),
Poisson brackets,
equation of motion,
spin frame impulses
respectively given by

\begin{align}
% \left\{
\renewcommand{\arraystretch}{1.2}
\begin{array}{rl}
\{
\rambda_{I\da}
\rambda_{\smash{J\db}\vphantom{\b}}
\}
\cdot
\,,\\
\{
\rambda_{I\da}
\lambda_\b{}^J
\}
\cdot
(\rambda^{-1})^{\dd J}
\,,\\
\{
\lambda_\a{}^I
\lambda_\b{}^J
\}
\cdot
(\rambda^{-1})^{\dc I}\hhnem
(\rambda^{-1})^{\dd J}
\,,
% \adjustbox{raise=-2ex}{\vphantom{0}}
\end{array}}
% \right.
\\[0.3ex]
% \left\{
\renewcommand{\arraystretch}{1.2}
\begin{array}{rl}
m_0\hem \dot{\rambda}_{I\da}
\rambda_{\smash{I\db}\vphantom{\b}}\hem
\,,\\
m_0\hem \dot{\lambda}_\a{}^I
\hem\lambda_\b{}^I}
% p_{\a\da}\mem (^{1+2}\hnem\omega')^\da{}_\db (p^{-1})^{\db\b}
% \hem\lambda_\b{}^I
\lambda_\a{}^J\, \rambda_{J\da}\mem (^{1+2}\hnem\omega')^\da{}_\db\mem (\rambda^{-1})^{\db I}
% \,.
\,,
% \adjustbox{raise=-2.05ex}{\vphantom{0}}
\end{array}}
% \right.
\\[0.38ex]
% \left\{
\renewcommand{\arraystretch}{1.2}
\begin{array}{rl}
\lsq\two_I|
\lsq\one_I|
\,+\,
i\hem (e_h\hem m_0 x^h \hem\mathe^{\three\cdot a})\,
\deltabar(2\mem\three{\kern0.02em\cdot\kern0.02em}\one)\,\mem
\lsq\bone_I\bthree\rsq\lsq\bthree|
\,,\\
\,+\,
i\hem (e_h\hem m_0 x^h \hem\mathe^{\three\cdot a})\,
\deltabar(2\mem\three{\kern0.02em\cdot\kern0.02em}\one)\,
{\textstyle\frac{x}{m_0}} \hem|\three\rangle\langle\three \one^I\rangle
% \,.
% \,,
\end{array}}
% \right.
\end{align}

in a self-dual plane wave background
with spinor-helicity variables $\three_\a$ and $\bthree_\da$.

has emphasized
the power and elegance of complex geometry [91, 94, 93, 89, 178, 179].
The central message of
our “zig-zag”

is that the physics of interacting massive spinning particles
should be understood in a way that
fully appreciates and respects
the complex-geometrical structure
that is already inherent in the free theory.
everything in the holomorphic/anti-holomorphic
the formulation of classical physics
more consistent with the
amplitudes-level understanding,

provided new geometrical insights on the physics of spin precession,
revealed properties

that were obscure in the conventional approach,

motivated a new way of doing calculations.

A few comments are in order.
It was

crucial in our analysis
that we have complexified the massive twistor space
to the massive ambitwistor space.
In particular,
for the heavenly equation of motion
to admit a solution,
it is necessary to “unlink”
$\lambda_\a{}^I$ with $\rambda_{I\da}$
and $z^{\da\a}$ with $\bz^{\da\a}$:
the scattering kinematics is complexified,
and both

spacetime coordinates $x^{\da\a}$ and
spin length $y^{\da\a}$ become complex variables.
the plane-wave scattering test in definite-helicity plane-wave backgrounds
considered in Section <ref>
will make no sense.

We also clarify
the role of pure self-duality in our discussion.

There seems
to be a fundamental obstruction
in having an exact complex-worldline description
in earthly backgrounds,
which can be traced back to the
“riddle” of Kerr black hole:
the existence of the opposite-helicity contact vertex [81].
As we elaborate further in [83],
all-order exact
Newman-Janis shift
of the particle's symplectic structure

manifests only
in heavenly backgrounds.
To pursue
a complex-worldline description
in the earthly setting,
it seems that
the background has to be linearized/abelianized.

For instance,
[2] was able to explore the Newman-Janis property
in the earthly setting
by taking the two-potential approach
to self-dual and anti-self-dual field strengths
at the linearized level.
In this paper, we have put our focus on the heavenly case

to discuss exact results.

There are still a lot more topics to explore.
For instance,
the exact solutions in Section <ref>
can also be obtained for arbitrary wave profiles.
It would be
more inspiring
if we could
discuss the memory effect

[180, 181, 182, 183, 184, 185]
and asymptotic symmetries of heavenly geometries [186]
in such a setting
and, in turn,
give a clearer physical interpretation

the impulses.
we could consider shockwave geometries
[187, 188, 189]
and provide a symplectic perturbation theory reincarnation of the Hamiltonian-based approach of

[93, 120].
Going further, one can relax various assumptions taken in our discussion.
For example,
one can go beyond three-point amplitudes

test-particle limit.
one can go beyond the equal-mass sector and

incorporate more general interactions
such as torque effects [53].

Working in terms of symplectic perturbations
might help
classifying such interactions

Finally, let us end with a few
suggestions for future directions.

Quadratic order in fields

Regarding the zig-zag expansions,
we have consistently ignored
terms from the second order.
we could have computed “four-point” zig-zag diagrams
as well.
For instance, suppose
a complexified background
$\omega' = \omega'^{\hem\four} + \omega'^{\hem\three}$
with both
self-dual and anti-self-dual modes
such that

\begin{align}
\label{eq:Compton-scheme}
% \omega' = \omega'_{\four} + \omega'_\three
% \,,\quad
\omega'^{\hem\four} =
\frac{i\nu_\four}{2}\mem
\bfour_\da \bfour_\db\mem
(dz \swedge dz)^{\da\db}
\,
\mathe^{-i\hem \bfour z \four}
\,,\quad
\omega'^{\hem\three} =
\frac{i\bar{\nu}_\three}{2}\mem
\three_\a \three_\b\mem
(d\bz \swedge d\bz)^{\a\b}
\,
\mathe^{-i\hem \bthree \bz \three}
\,.
\end{align}

we can consider zig-zag vector fields such as

\begin{align}
% \begin{split}
%     \label{eq:4vec+-}
%     \smash{
%         \rlap{\makebox[20.00pt][l]{}\adjustbox{raise=15.760pt,scale=0.85}{$\four$}}
%         \rlap{\makebox[39.24pt][l]{}\adjustbox{raise=-16.515pt,scale=0.85}{$\three$}}
%         \includegraphics[valign=c]{figs/l-b.pdf}
%         \includegraphics[valign=c]{figs/v+.pdf}
%         \includegraphics[valign=c]{figs/v-.pdf}
%         \includegraphics[valign=c]{figs/xD.pdf}
%     }
%     &= \{\blank,z^{\da\a}\}^\circ\,
%     \frac{8i}{\bDelta}\,
%     % (-8i/\bDelta)\,
%     \omega'^{\hem\four}_{\smash{\da\db}\vphantom{\b}}(z)
%     \, y^{\db\b} \,
%     \bomega'^{\hem\three}_{\b\a}(\bz)
%     \,,\\
%     &= \{\blank,z^{\da\a}\}^\circ\,\mem
%     i
%     \Big(
%         {-2\nu_\four\bar{\nu}_\three\mem [\bfour|y|\three\rangle}
%         \mem
%         (-\three_\a\bfour_\da)
%         \hnem
%     \Big)
%     \,e^{-i \bthree \bz \three -i\bfour z\four}
%     /\bDelta
%     \,,
%     % \,.
% \end{split}
%     \\
\begin{split}
\label{eq:4vec-+}
\smash{
\rlap{\makebox[39.24pt][l]{}\adjustbox{raise=15.760pt,scale=0.85}{$\four$}}
\rlap{\makebox[20.00pt][l]{}\adjustbox{raise=-16.515pt,scale=0.85}{$\three$}}
\includegraphics[valign=c]{figs/l.pdf}
\includegraphics[valign=c]{figs/v-.pdf}
\includegraphics[valign=c]{figs/v+.pdf}
\includegraphics[valign=c]{figs/xD-b.pdf}
&= \{\blank,\bz^{\da\a}\}^\circ\,
\frac{-8i}{\Delta}\,
% (+8i/\Delta)\,
\bomega'^{\hem\three}_{\a\b}(\bz)
\, y^{\db\b} \,
\omega'^{\hem\four}_{\smash{\db\da}\vphantom{\b}}(z)
\,,\\
&= \{\blank,\bz^{\da\a}\}^\circ\,\mem
\Big(\hem\hhhem
{2\nu_\four\bar{\nu}_\three\mem \lsq\bfour|y|\three\rangle}
\mem
\hnem
\Big)
\,e^{-i\hem \bthree \bz \three -i\hem \bfour z\four}
% \,.
\,,
\end{split}
\end{align}

which contributes to the impulse at the quadratic order.

the combinations $\lsq\bfour|y|\three\rangle$
and $(-\three_\a\bfour_\da)$
when studying the notorious Compton amplitude of Kerr
we can
speculate on
a relationship between
zig-zag diagrams such as (<ref>)
and the
four-point amplitudes.
Higher-order zig-zag diagrams indeed seem to be
a unique feature of the spinning particle,
as for the spin-less case
$\omega'^{\hem\four}$ and $\omega'^{\hem\three}$
cannot cascade
because $\{x^{\da\a},x^{\db\b}\}^\circ = 0$.
it remains to be
whether symplectic perturbations can be directly related to amplitudes
at higher orders;
as mentioned in [82] as well,
we only navely expect that
a “dequantization” of the S-matrix will be
a time-ordered action of the deformed Hamiltonian vector fields
as infinitesimal diffeomorphisms on the physical phase space,
à la geometric prequantization [190].
further $\mathcal{O}(\phi^2)$
terms can
in principle
enter into the symplectic perturbation
when there are both self-dual and anti-self-dual modes.

Overall, whether zig-zag diagrams will also be successful in illuminating Kerr's four-point physics needs further investigation.

Physical interpretation of zig-zag diagrams

Now we would like to ask
bigger questions.
Let us
remark on the inspirations behind
the zig-zag diagram notation
and envision further developments of the idea.

Zig-zag diagrams started from denoting
$\{z^{\da\a},\bz^{\db\b}\}^\circ =$ $-2iy^{\da\b} (\rambda^{-1}\nem\lambda^{-1})^{\db\a}$ and $\phi^\pm_{mn}$
in the Penrose graphical notation
[87, 88, 89, 90]:

\begin{align}
\begin{split}
\label{eq:Penrosegraphical}
\{z^{\da\a},\bz^{\db\b}\}^\circ =
\,
\includegraphics[valign=c,scale=1.2]{figs/PB-z-zbar.pdf}
\{\bz^{\da\a},z^{\db\b}\}^\circ =
\,
\includegraphics[valign=c,scale=1.2]{figs/PB-zbar-z.pdf}
\,,\\
\phi^+_{mn}\hnem(z)\mem
\e_{\a\b}\hspace{0.12em}(\bs^{mn})_{\smash{\da\db}\vphantom{\b}} =
\includegraphics[valign=c,scale=1.2]{figs/PB+.pdf}
\phi^- _{mn}\hnem(\bz)\hem
(\s^{mn})_{\a\b}\mem\be_{\smash{\da\db}\vphantom{\b}} =
\includegraphics[valign=c,scale=1.2]{figs/PB-.pdf}
\,.
% \,,
\end{split}
\end{align}

evolved to
the notations

one can regard
the double-line format
of zig-zag diagrams
as representing
the $\mathrm{SL}(2,\mathbb{C})$ index flows.

such a double-line

(or “ribbon”)
design is reminiscent of
the “worldsheet” of
Guevara, Maybee, Ochirov, O'Connell, and Vines

In fact, the reader might have noticed that
we pretended as if the white/black filling
somehow represents an actual surface connecting
$\s {\,\mapsto\,} z^m\hnem(\s)$ and $\s {\,\mapsto\,} \bz^m\hnem(\s)$
(which may be called
“GMOOV worldsheet” [2]):

\begin{align}
\omega'^+
\,\,\,\,=\,\,\,\,
\left\{\,\,\mem
\renewcommand{\arraystretch}{1.5}
\begin{array}{cccccccl}
\includegraphics[valign=c,scale=1.0]{figs/insert+.pdf}
& &
& &
& &
\,,
\\
\includegraphics[valign=c,scale=1.0]{figs/insertg+ww.pdf}
\includegraphics[valign=c,scale=1.0]{figs/insertg+wb.pdf}
\includegraphics[valign=c,scale=1.0]{figs/insertg+bw.pdf}
\includegraphics[valign=c,scale=1.0]{figs/insertg+bb.pdf}
&\,\,\,\text{(``exponential arbitrary-$g$'')}
\,,
\\
\includegraphics[valign=c,scale=1.0]{figs/insert0+ww.pdf}
\includegraphics[valign=c,scale=1.0]{figs/insert0+wb.pdf}
\includegraphics[valign=c,scale=1.0]{figs/insert0+bw.pdf}
\includegraphics[valign=c,scale=1.0]{figs/insert0+bb.pdf}
&\,\,\,\text{(maximally non-minimal)}
\,,
\\
& &
& &
& &
\includegraphics[valign=c,scale=1.0]{figs/insertA+.pdf}
&\,\,\,\text{(antipodal minimal)}
\,.
\end{array}
\right.
\end{align}

The dot can be thought of as indicating the incident point of a self-dual “non-linear quantum.”
The coupling is minimal
when the incident point lies
on the holomorphic worldline.

Thus, thinking of the “graphical realism” of Feynman diagrams,[
a natural graphical representation of a perturbative series is believed to represent

one might ask,
“do zig-zag diagrams represent an actual physical

entity, such as the worldsheet?”

It seems too early to answer such philosophical questions.
the association of
zig-zag vector fields with amplitudes
such as
in (<ref>) or (<ref>)
suggests that

the graphical elements of zig-zag diagrams
might be
modules for building up various

“vertices” for

spinning particle.

At least,
we should stress that
zig-zag diagrams do not represent

spacetime trajectories
of a massive spinning particle,
although we
made an analogy between zig-zag and Feynman diagrams
by imagining $\omega^\circ{}^{-1}$ as “propagators”
and $\omega'$ as “vertices.”
Our zig-zag theory
is not the same as Penrose's original concept of the zig-zag electron;
instead, it is a certain remake of it.

The zigs and zags
referred to
holomorphic and anti-holomorphic objects in the massive twistor space,
which in turn get associated with

left and right
chiralities of the background field that the particle interacts with.

If one wants to associate

the image of “zig-zag” with spacetime trajectories,

can imagine a minimally coupled massive twistor particle
moving in an earthly background
as depicted in
Figure <ref>
(while ignoring the contact vertices).

Penrose's zig-zag electron, reimagined.
(Cf. Fig. 25.1b of [91].)

Massive twistor diagrams?

Our explorations
on the
the massive twistor particle

extend to
Hodges diagram
[92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105],
which was indeed another inspiration
behind the development of zig-zag diagrams.

Let us reimagine zig-zag diagrams
in a Hodges-like

The implementation

is again by Penrose graphical notation,
but with
the “differentiation balloon” of Penrose

as well as
birdtracks/trace diagram notations
for the $\mathrm{SU}(2)$ epsilon tensor
(see [90] and references therein).
The basic elements are

\begin{align}
\includegraphics[valign=c,scale=1.0]{figs/hodges/dotZind.pdf}
\,,\quad
\bZ_I{}^\rmA
\includegraphics[valign=c,scale=1.0]{figs/hodges/dotZind-b.pdf}
\,,\quad
\e_{IJ} = \includegraphics[valign=c,scale=1.0]{figs/hodges/edown.pdf}
\,,\quad
\e^{IJ} = \includegraphics[valign=c,scale=1.0]{figs/hodges/eup.pdf}
\,.
\end{align}

The difference in thickness distinguishes

the vector spaces.
Also, let us denote
differential forms by “highlighted” dots:

\begin{align}
\includegraphics[valign=c,scale=1.0]{figs/hodges/dZind.pdf}
\,,\quad
\includegraphics[valign=c,scale=1.0]{figs/hodges/dZind-b.pdf}
\,.
\end{align}

The free theory Poisson bivector $\omega^\circ{}^{-1}$
is suggestive of the Fourier transform

between the twistor and dual twistor spaces,
which is drawn like squiggly “propagators” in the Hodges notation.
In this light, we denote

\begin{align}
\,\,\,\,\text{or}\,\,\,\,
\quad\rightsquigarrow\quad
\includegraphics[valign=c]{figs/hodges/PBsquig.pdf}
\end{align}

so that

\begin{align}
\frac{\partial}{\partial Z_\rmA{}^I}
\includegraphics[valign=c]{figs/hodges/partialZind-b.pdf}
\,,\qquad
\frac{\partial}{\partial\bZ_I{}^\rmA}
\includegraphics[valign=c]{figs/hodges/partialZind.pdf}
\,,\qquad\,
\includegraphics[valign=c]{figs/hodges/PBid-1.pdf}
\includegraphics[valign=c]{figs/hodges/PBid-2.pdf}
% =
% \delta_I{}^J \delta_\rmA{}^\rmB
\,.
\end{align}

Now consider the self-dual plane wave symplectic perturbation for electromagnetism:

\begin{align}
\omega'^+
\mem=\mem
% =
% \frac{iq}{\sqrt{2}}\mem \bo_\da \bo_\db
% (dz \swedge dz)^{\da\db}
% \,\mathe^{-i\bozo}
% =
\hem(dZ_\rmA{}^I\nem\nem\wedge dZ_\rmB{}^J)\mem
\e_{IJ}
\,\mathe^{-i\hem\bthree{z}\three}
\,.
\end{align}

We can half-Fourier transform this into the photon's dual twistor space,
$\bW^A = \big( -i\bar{\varpi}^\a \,\,\mem \bthree_\da \big)$:

\begin{align}
\label{eq:disc.MTem-2}
\omega'^+
\mem=\mem
(\bW^\rmA dZ_\rmA{}^I) {\mem\wedge\mem} (\bW^\rmB dZ_\rmB{}^J)\mem
\e_{IJ}
\,\delta^{(2)}\hnem(\bar{\varpi}-\bthree z)
\,.
\end{align}

the photon's massless dual twistor is incident at the massive particle's holomorphic position $z^{\da\a}$.
This hints that the photon's propagator is also “zig-zag”
in the field theory of Newman-Janis shifted fields in spin-space-time:
Zwanziger electromagnetism
[191, 192, 193, 194, 195].

if the

delta function is dropped
for simplicity,
(<ref>) appears as

\begin{align}
\omega'^+
\mem=\mem
\includegraphics[valign=c]{figs/hodges/Oem-1.pdf}
\,.
\end{align}

In turn,
the vector field
\includegraphics[valign=c]{figs/v+.pdf}
\includegraphics[valign=c]{figs/xD-b.pdf}$
is graphically computed as

\begin{align}
\begin{split}
\frac{i\sqrt{2}q}{\Delta}\,
\includegraphics[valign=c]{figs/hodges/Oem-calc1.pdf}
\,=\,\mem
\frac{i\sqrt{2}q}{\Delta}\,
\includegraphics[valign=c]{figs/hodges/Oem-calc2.pdf}
\,.
\end{split}
\end{align}

Following [92],
the infinity twistors are denoted as dashed lines.

While Guevara [14] has half-Fourier transformed the classical-spin amplitudes
to the twistor space of the massless particle,
it is tempting to have

a description of the amplitudes
that is
fully twistorial
for both massless and massive particles.
In this context,
it would be interesting to see if
half-Fourier transformed symplectic perturbations and zig-zag vector fields

can lead to
a Hodges diagram for massive particles.

fifty years,
twistor theory has provided
a unique perspective
on the physics of
massless particles and fields in four dimensions.
We hope
our zig-zag symplectic perturbation theory
in the massive twistor space
could signal the dawn of
“massive twistor theory.”

We are grateful to
Clifford Cheung,
Julio Parra-Martinez,

Andreas Helset,

Jung-Wook Kim
for discussions and comments on the manuscript.
JHK would like to thank
Clifford Cheung,
Julio Parra-Martinez,
Andreas Helset,
Jordan Wilson-Gerow,
Zander Moss
enlightening conversations and
helpful discussions.
The work of JHK is supported
in part by Ilju Academy and Culture Foundation.

The work of SL is supported in part by
the National Research Foundation of Korea grant NRF-2019R1A2C2084608.

§ TBMT AND MPTD EQUATIONS

The Thomas-Bargmann-Michel-Telegdi equation

[41, 42, 43, 44, 45, 46]
describes a spinning particle moving in a background electromagnetic field
in special relativity.
In this paper,
by a massive spinning particle is governed by the TBMT equation with a certain $g$ value
we mean that its momentum and spin length pseudovector
obey the following equations
in the covariant spin gauge with the gauge fixing $-p_m\hem y^m = 0$ of the residual redundancy:

\begin{align}
\begin{split}
\label{eq:BMT}
\dot{p}_m
+ p_n\mem \phi^n{}_m/m
% + \mathcal{O}(y^1\phi^1,\phi^2)
+ \mathcal{O}(y^1)
\,,\\
\dot{y}^m
- \frac{1}{m} \bigg(\mem{
\frac{g}{2}\, \phi^m{}_n
- \Big(1{\mem-\mem}\frac{g}{2}\mem\Big)\,
\frac{\hhem{p^m p_l}\hhem}{m^2}\, \phi^l{}_n
\mem y^n
% + \mathcal{O}(y^1\phi^1,\phi^2)
+ \mathcal{O}(y^2)
\,.
\end{split}
\end{align}

$\phi_{mn}$ is an antisymmetric tensor.

We use these terminologies
“TBMT equation” and “$g$ factor”
in a broad, general sense
that does not restrict
to be the field strength of electromagnetism.
For instance,
it can be something depending on other dynamical variables of the spinning particle
such as $q_a F^a{}_{mn}\hnem(x)$ of Yang-Mills
or $2\hem p_r\gamma^r{}_{[mn]}\hnem(x)$ of gravity.

The Mathisson-Papapetrou-Tulczyjew-Dixon equation [47, 48, 49, 50, 51, 52, 53, 54]
the dynamics of
a spinning particle in curved spacetime.
It is extended by a parameter $\kappa$,
which is
called the  “gravimagnetic ratio”
in the literature
[55, 56, 57, 58, 59, 60, 61].
Our version

\begin{align}
\begin{split}
\label{eq:MPD}
m\mem e^m{}_\m\hnem(x)\mem \dot{x}^\m
- p^m
+ \mathcal{O}(y^1)
\,,\\
\dot{p}_m
- p_n\mem \gamma^n{}_{ms}\hnem(x)\mem e^s{}_\s\hnem(x)\mem \dot{x}^\s
% - p_n\mem \gamma^n{}_{m\s}\hnem(x)\mem \dot{x}^\s
- \frac{g}{2}\, p_n\mem {\star R}^n{}_{mrs}\hnem(x)\mem y^r\hem e^s{}_\s\hnem(x)\mem \dot{x}^\s
+ \mathcal{O}(y^2)
\,,\\
\dot{y}^m
+ \bigg(\mem{
\frac{g}{2}\, \gamma^m{}_{ns}\hnem(x)
- \Big(1{\mem-\mem}\frac{g}{2}\mem\Big)\,
\frac{\hhem{p^m p_l}\hhem}{m^2}\, \gamma^l{}_{ns}\hnem(x)
\mem y^n\mem e^s{}_\s\hnem(x)\mem \dot{x}^\s
\\
\dot{y}^m
+ \bigg(\mem{
\kappa\, {\star R}^m{}_{nrs}\hnem(x)
- \Big(\hem\frac{g}{2}{\mem-\mem}\kappa\Big)\,
\frac{\hhem{p^m p_l}\hhem}{m^2}\, {\star R}^l{}_{nrs}\hnem(x)
\mem y^n\mem y^r\mem e^s{}_\s\hnem(x)\mem \dot{x}^\s
+ \mathcal{O}(y^3)
\,,
% \,.
\end{split}
\end{align}

with the covariant gauge and
the constraint
$-p_m\hem y^m = 0$.
denote the
spin connection coefficients,
and Riemann tensor.
As in (<ref>),
the momentum equation

the “zero point”
of non-minimal parameters:
the precessions of $p_m$ and $y^m$ are synchronized when

$1 = g/2 = \kappa$.
By including $g$ in (<ref>), we are being maximally pedantic,
intending to maximally mimic

the amplitude.

Of course, it is fixed to $g{\,=\,}2$ by
the general covariance of the spin vector:
the spin vector should behave as the same as
two infinitesimally deviated points.
For more details,
see Appendix B of [9].

Either of the two bracketed terms in the $\dot{y}^m$ equation
can be viewed as a “double copy” of the bracketed term in (<ref>).
This is because there exist two ways of drawing analogies between electromagnetism/Yang-Mills and gravity.
The first term,
which defines the “gravitational gyromagnetic ratio” $g$,
follows from (<ref>) by
$\phi_{mn} = -m\mem \gamma_{mn\r}\hnem(x)\mem \dot{x}^\r
\,(\sim p_r\hem 2\gamma^r{}_{[mn]}\hnem(x))$.
Then one is comparing $A$ to $e^m$ and $F$ to $\gamma^m{}_n$
as in Kaluza-Klein reduction [196, 197, 198, 199, 200] or the gravitoelectromagnetism formulation of linearized gravity [201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218],
which realizes the “connection as field strength” argument in [82].
This parallel is suggested from
the usual formulation of flat-space graviton scattering amplitudes in terms of metric perturbations
and also from
the “exact sequences of electromagnetism and gravity” diagram in [82].

More poetically,
we could say
“Aharanov-Bohm double copies to Sagnac” [219]:
the Coriolis force is the gravitomagnetic force.
On the other hand, the second term follows from (<ref>) by
$\phi_{mn} = -m \,{\star R}_{mnrs}\hnem(x)\mem y^r\hem e^s{}_\s\hnem(x)\mem \dot{x}^\s$
putting $g {\mem=\mem} 2$
and replacing $g/2$ with $\kappa$.
one is comparing $A$ to $\gamma^m{}_n$ and $F$ to $R^m{}_n$.

This way of thinking is more natural in a background independent/non-perturbative set-up
and is encouraged by
Ashtekar variables [220, 221, 222, 223, 224, 225, 226],
non-linear graviton [157, 156], Weyl double copy [227, 228, 229, 230],
covariant color-kinematics duality [231],

Both of these two parallels
emerge in our discussion on
symplectic perturbation theory
of spinning particles:
the former from the non-covariant scheme
employed in this paper
and the latter from the covariant scheme
employed in the follow-up paper [83].

Generic O(ϕ¹) spin precession equation of motion

In this paper, we have proposed a generic spin precession equation of motion, all orders in spin and first order in the formal “field strength” $\phi$.
When $\phi$ is self-dual, it reads

\begin{align}
\label{eq:prec-eq.vec}
\begin{split}
\dot{p}_m
+ \frac{1}{m}\,
\Bigg[\mem{
\sum_{\ell=0}^{\infty}\mem
\frac{
(C_\ell {\mem+\mem} \tilde{C}_\ell \mem\star^{-1})
\big({
(y{\mem\cdot\mem}\partial)^\ell\mem {\star^\ell\hnem \phi}(x)
\mem+\mem \mathcal{O}(\phi^2)
\,,\\
\dot{y}^m
- \frac{1}{m}\nem\hnem
\left[{
\nem
\begin{array}{l}
\displaystyle
\sum_{\ell=0}^{\infty}
\frac{
(C_{\ell+1} {\mem+\mem} \tilde{C}_{\ell+1} \mem\star^{-1})
\big({
(y{\mem\cdot\mem}\partial)^\ell\mem {\star^\ell\hnem \phi}(x)
\\
\displaystyle
\frac{p^mp_r}{m^2}\mem
\sum_{\ell=0}^{\infty}
\frac{
(C_\ell {\mem+\mem} \tilde{C}_\ell \mem\star^{-1})
(C_{\ell+1} {\mem+\mem} \tilde{C}_{\ell+1} \mem\star^{-1})
\big({
(y{\mem\cdot\mem}\partial)^\ell\mem {\star^\ell\hnem \phi}(x)
\end{array}
\hhnem\hnem\nem
\\
&\quad \mem+\mem \mathcal{O}(\phi^2)
% \,.
\,,
\end{split}
\end{align}

we have also included magnetic Wilson coefficients
for full generality.

One obtains
(<ref>) and (<ref>)
by truncating (<ref>)
with having
$C_1 {\,=\,} g/2$, $C_2 {\,=\,} \kappa$,
and $\tilde{C}_\ell {\,=\,} 0$.

As pointed out recently in [2],
$\dot{p}_m$ and $\dot{y}^m$
can be described together in a more unified manner
in terms of the spin frames
as $\dot{\lambda}{}_\a{}^I$, $\dot{\rambda}{}_{I\da}$.[
This idea can be traced back to
the works [123, 232, 233, 234].
This is best illustrated
the body-frame components

of the spin

are constant,
which happens in our setting
the particle has a constant Regge trajectory.
The spin frame version of (<ref>) reads

\begin{align}
\begin{split}
\label{eq:prec-eq.frames}
m_0\mem \dot{\rambda}{}_{I\da}
\rambda_{\smash{I\db}\vphantom{I\da}}
\mem
\Bigg[\mem{
\sum_{\ell=0}^{\infty}\mem
\frac{1}{\hhem\ell!\hhem}
\frac{
\phi^\db{}_\da\hnem(x)
\mem+\mem \mathcal{O}(\phi^2)
\,,\\
m_0\mem \dot{\lambda}{}_\a{}^I
\lambda_\a{}^J\mem
\mem
\rambda_{\smash{J\db}\vphantom{I\da}}
\Bigg[\mem{
\sum_{\ell=0}^{\infty}\mem
\frac{1}{\hhem\ell!\hhem}
\frac{
\phi^\db{}_\da\hnem(x)
(\rambda^{-1})^{\da I}
\mem+\mem \mathcal{O}(\phi^2)
% \,.
\,,
\end{split}
\end{align}

where $C^\pm_\ell = C_\ell {\,\mp\,} i\hem \tilde{C}_\ell$.

One can clearly see that
$\dot{\lambda}{}_\a{}^I = 0 + \mathcal{O}(\phi^2)$
in self-dual backgrounds
implies $C^+_\ell {\,=\,} C^+_0 {\,=\,} 1$
and vice versa:
all three-point spin multipole couplings
(not only dipole and quadrupole, to emphasize)
to the self-dual field
are Kerr-Newman.[

We clarify that
by “Kerr-Newman coupling”
we mean $C^\pm_\ell {\,=\,} 1$
for both helicities $1$ and $2$.
The electromagnetic coupling of the Kerr-Newman black hole coincides with
the  coupling [19].

These equations
implied throughout
(<ref>), (<ref>), (<ref>), (<ref>), (<ref>),

we have
in terms of
the real-worldline Wilson coefficients
for reference.

Lastly, we spell out
the spin frame versions of
(<ref>) and (<ref>):

\begin{align}
\begin{split}
\label{eq:frames-BMT}
\dot{\rambda}{}_{I\da}
&{}\mem=\mem \textstyle
\frac{1+g/2}{2}\,\hem
\rambda_{\smash{I\db}\vphantom{I\da}}\mem\mem \phi^\db{}_\da\hnem(x)
+ \mathcal{O}(y^1)
\,,\\
\dot{\lambda}{}_\a{}^I
&{}\mem=\mem \textstyle
\frac{1-g/2}{2}\,\hem
\lambda_\a{}^J\,
\rambda_{\smash{J\db}\vphantom{I\da}}\mem
\phi^\db{}_\da\hnem(x)\hhem
(\rambda^{-1})^{\da I}
\nem\hnem
+ \mathcal{O}(y^1)
\,,
% \vphantom{\Big]}
\end{split}\\[0.2ex]
\begin{split}
\label{eq:frames-MPD}
\dot{\rambda}{}_{I\da}
&{}\mem=\mem \textstyle
\rambda_{\smash{I\db}\vphantom{I\da}}\mem
% \mem\minie\hnem
\Big[\hem{
\frac{1+g/2}{2}\mem\hhem
\gamma^\db{}_{\da s}\hnem(x)
\frac{g/2+\kappa}{2}\hem
R^\db{}_{\da rs}\hnem(x)\mem
+ \mathcal{O}(y^2)
\, e^s{}_\s\hnem(x)\mem \dot{x}^\s
\,,\\[-0.2ex]
\dot{\lambda}{}_\a{}^I
&{}\mem=\mem \textstyle
\lambda_\a{}^J\,
\rambda_{\smash{J\db}\vphantom{I\da}}\mem
% \mem\minie\hnem
\Big[\hem{
\frac{1-g/2}{2}\mem\hhem
\gamma^\db{}_{\da s}\hnem(x)
\frac{g/2-\kappa}{2}\hem
R^\db{}_{\da rs}\hnem(x)\mem
+ \mathcal{O}(y^2)
(\rambda^{-1})^{\da I}
\, e^s{}_\s\hnem(x)\mem \dot{x}^\s
\,,
% \nonumber
\end{split}
\end{align}

§ MATCHING CALCULATION

Classical spin from coherent states

Aoude and Ochirov
[10]
explicated the procedure of obtaining massive spinning amplitudes in the classical spin limit
in terms of spin coherent states.
We briefly review their methodology in our notation.
See also [17] for a succinct review.

Modeling the spinning degrees of freedom
by the coadjoint orbit [235, 236, 237, 238, 239],
the classical value of spin angular momentum is identified with its coherent-state expectation value in the first-quantized theory as

\begin{align}
\label{eq:psiW}
W_a = (\hbar/2)\mem \bpsi^I (\s_a)_I{}^J \psi_J
\,,\quad
% W_0 = -\minie\mem \bpsi^I \psi_I = -|\vec{W}|
|\vec{W}| = (\hbar/2)\mem \bpsi^I \psi_I
\,.
\end{align}

The classical-spin amplitude then follows from the
expectation value of all definite-spin amplitudes in the $\hbar\to0$ limit with $|\vec{W}|$ fixed:

\begin{align}
\label{eq:Aoudeformula}
\begin{split}
% \label{eq:Aoudeformula-1}
\mathcal{M}\hnem(\one,a;\three)
{\phantom{{}:={}}\mathllap{{}={}}}{}& \lim_{\substack{\hbar\to0 \\ \bpsi\psi = 2|\vec{W}|/\hbar}}
\sum_{2s=0}^\infty
\mathcal{M}_{(2s)}\hnem(\one,\bpsi,\psi;\three)
\,,\\
% \label{eq:Aoudeformula-2}
\mathcal{M}_{(2s)}\hnem(\one,\bpsi,\psi;\three)
\frac{\mathe^{-\bpsi\psi}}{(2s)!}\,\mem
\bpsi^{J_1}\cdots\bpsi^{J_{2s}}
\mem
(\mathcal{A}_{(2s)}\hnem(\two{\leftarrow}\one;\three))_{J_1\cdots J_{2s}}{}^{I_1\cdots I_{2s}}
\,
\psi_{I_1}\cdots\psi_{I_{2s}}
\,,
\end{split}
\end{align}

where $\mathe^{-\bpsi\psi}$ is the measure normalizing coherent states.
For example, the spin-tensor $\identity^{\nem\odot 2s}$ of the plus-helicity minimal amplitude
\mathcal{A}^{+h}_{(2s)}\hnem(\two{\leftarrow}\one;\three)
= e_h\hem m_0 x^h\mem \identity^{\nem\odot 2s}
boils down to

\begin{align}
\begin{split}
\label{eq:expfactor}
% \text{\eqref{eq:3kin-real}}
% \qiq
% \lim_{\bpsi\psi \to \infty}
\mathe^{-\bpsi\psi}
\exp\nem\Big(
\bpsi^J\nem\hnem\langle\two^{-1}_J\one^I\rangle\psi_I
\nem\Big)
% = \lim_{\bpsi\psi \to \infty} e^{\three\cdot a}
\mem=\mem \mathe^{\three\cdot a}
\,,
\end{split}
\end{align}

upon plugging in the real on-shell kinematics given in (<ref>)
[7, 10].
Note that

\begin{align}
\begin{split}
\label{eq:3a-identity}
\langle\two^{-1}_J\one^I\rangle
&= \delta_J{}^I
- \minie\hem \langle\one^{-1}_I\three\rangle \lsq\bthree\bone^{-1}{}^J\rsq
\,,\\
&= \delta_J{}^I
+ \three_m
\Big(\hnem{
\textstyle
\frac{\hbar}{2m_0}\hem (\L(\one))^m{}_a\hem (\s^a)_I{}^J
+ \frac{1}{2}\hem (\one^{-1})^m\hem \delta_I{}^J
\,,\\
&= \delta_J{}^I
\three_m (\L(\one))^m{}_a\hem (\s^a)_I{}^J
% \,,\\
% \iq
% \bpsi \langle\two^{-1}\one\rangle \psi - \bpsi\psi
% &= \three \cdot a(\one) / \hbar
% \,.
\end{split}
\end{align}

on the support of $\deltabar(2\mem\one{\mem\cdot\mem}\three)$.
In turn, it follows that

\begin{align}
\bpsi^J \langle\two^{-1}_J\one^I\rangle \psi_I - \bpsi^I\psi_I
\mem=\mem
\three{\mem\cdot\mem} a(\one)
\,.
\end{align}

In our spherical top setting,
such classical-spin states
are realized in the $(\lambda,\rambda)$ polarization as
polynomials of $\lambda_\a{}^I$ and $(\rambda^{-1})^{\da I}$
saturated by a common spinor $\psi_I$;
i.e., they are the “right” coherent states with respect to the $\mathrm{SU}(2)$ group action on $\mathbb{MA}$.
The unoccupied “left” $\mathrm{SL}(2,\mathbb{C})$ indices contract to amplitudes as spacetime spinor-tensors.

Applying (<ref>) with the real kinematics (<ref>),

the coherent-state expectation value of the plus-helicity spinning amplitude (<ref>)
is given as

\begin{align}
\begin{split}
% \label{eq:M+2scontrib}
% \mathcal{M}^+_{(2s)}\hnem[g^+](\psi;\bpsi;\three)
% &= e_h\hem m_0 x^h
% \mem\mathe^{-\bpsi\psi}
% \mem\sum_{\ell=0}^{2s}\mem
% \frac{g^+_\ell}{(2s)!}\mem
% \big(\bpsi\langle \two^{-1} \one \rangle \psi\big)^{2s-\ell}
% \left(\hnem
%     \frac{x}{m_0}
%     \bpsi
%     \langle \two^{-1} \three\rangle
%     \langle \three \one \rangle
%     \psi
% \nem\right)^{\kern-0.23em\ell}
% \,,\\
% &= e_h\hem m_0 x^h
% \mem\mathe^{-\bpsi\psi}
% \mem\sum_{\ell=0}^{2s}\mem
% \frac{g^+_\ell}{(2s)!}\mem
% \big(\bpsi\psi + \three{\kern0.02em\cdot\kern0.02em}a \big)^{2s-\ell}
% (-2\mem\three{\kern0.02em\cdot\kern0.02em}a)^\ell
% \,,
\label{eq:M+2scontrib}
% & \mathcal{M}^{+h}\hnem[g^+](x,a;\three)
% \\
% &=
% \lim_{\bpsi\psi\to\infty}
e_h\hem m_0 x^h
\mem\mathe^{-\bpsi\psi}
\mem\sum_{2s=0}^{\infty} \sum_{\ell=0}^{2s}
\frac{\hem g^+_\ell \hnem \hbar^\ell \hem}{(2s)!}\mem
\big(\bpsi\psi + \three{\mem\cdot\mem}a\big)^{2s-\ell}
\,,\\
% \lim_{\bpsi\psi\to\infty}
e_h\hem m_0 x^h
\mem\mathe^{-\bpsi\psi}
\mem\sum_{\ell=0}^{\infty}
\mem g^+_\ell \hnem \hbar^\ell
\nem\hnem \left(
\frac{-2\mem\three{\mem\cdot\mem}a}{\vphantom{\adjustbox{raise=0.2em}{$\bpsi$}}\bpsi\psi + \three{\mem\cdot\mem}a}
\right)^{\kern-0.17em\ell}
\sum_{2s=\ell}^{\infty}
\frac{1}{(2s)!}\hem
\big(\bpsi\psi + \three{\mem\cdot\mem}a\big)^{2s}
\,,\\
% \lim_{\bpsi\psi\to\infty}
e_h\hem m_0 x^h
\mem\mathe^{\three\cdot a}
\mem\sum_{\ell=0}^{\infty}
\mem g^+_\ell \hnem \hbar^\ell
\nem\hnem \left(
\frac{-2\mem\three{\mem\cdot\mem}a}{\vphantom{\adjustbox{raise=0.2em}{$\bpsi$}}\bpsi\psi + \three{\mem\cdot\mem}a}
\right)^{\kern-0.17em\ell}
\left(\hnem
1 - \frac{1}{(\ell-1)!}\mem \Gamma\big(\ell\hem\big|\hem\bpsi\psi + \three{\mem\cdot\mem}a\big)
\nem\kern-0.015em\right)
% \,,
\,.
\end{split}
\end{align}

As argued in [10] around equations (3.22)-(3.27),
for large $\bpsi\psi$ this has the limit

\begin{align}
\begin{split}
\label{eq:M+coherent}
& \mathcal{M}^{+h}\hnem[g^+](x,a;\three)
e_h\hem m_0 x^h
\mem\mathe^{\three\cdot a}
\hnem
\left(
\lim_{\hbar\to0}
\mem\sum_{\ell=0}^{\infty}
\mem g^+_\ell \hnem \hbar^\ell\mem
\nem \bigg({
\,\right)
\,.
\end{split}
\end{align}

Therefore, provided that $g^+_\ell$ scales as $\mathcal{O}(\hbar^{-\ell})$,
we conclude that

\begin{align}
\label{eq:gen-fgrelation}
% f^+_\ell =
% \frac{1}{(-|\vec{W}|)^\ell}
% \lim_{\hbar\to0} g^+_\ell\hbar^\ell
= (-|\vec{W}|)^{-\ell} \cdot \lim_{\hbar\to0} g^+_\ell \hnem \hbar^\ell
\qiq
= g^+\hnem(-\xi/|\vec{W}|)
\end{align}

by comparing (<ref>) with
\mathcal{M}^{+h}\hnem(\one,a;\three)
\,{\cdot}\, f^+\hnem(\three{\mem\cdot\mem}a)$
of (<ref>).

A trick

Lastly, we demonstrate a slightly quicker derivation.
As shown in (<ref>),
the real kinematics (<ref>) has induced the minimal exponential spin factor $\mathe^{\three\cdot a}$.
However, we find that we can either apply the complexified kinematics (<ref>) and put $\mathe^{\three\cdot a}$ by hand.
For minimal coupling, it works as

\begin{align}
\begin{split}
% \text{\eqref{eq:3kin-plus}}
\one_\a{}^I \hnem=\mem \two_\a{}^I
\qiq
\M(\one,a;\three)
\mem&=\mem
e_h\hem m_0 x^h\mem
e^{\three\cdot a}
\left(
\mathe^{-\bpsi\psi}
\exp\nem\Big(
\bpsi^J\nem\hnem\langle\two^{-1}_J\one^I\rangle\psi_I
\nem\Big)
\right)
\,,\\
\mem&=\mem
e_h\hem m_0 x^h\mem
e^{\three\cdot a}
\,.
\end{split}
\end{align}

This trick may be thought of as
an analog of
taking the holomorphic complex worldline as the “zero point”:
Newman-Janis shifted
is taken as default.
One can check that it also works in the non-minimal case:

\begin{align}
\begin{split}
\mathe^{\three\cdot a}\left(
e_h\hem m_0 x^h
\mem\mathe^{-\bpsi\psi}
\mem\sum_{2s=0}^{\infty} \sum_{\ell=0}^{2s}
\frac{\hem g^+_\ell \hnem \hbar^\ell\hem}{(2s)!}\mem
\right)
\,,\\
\mathe^{\three\cdot a}\left(
e_h\hem m_0 x^h
\mem\sum_{\ell=0}^{\infty}
\mem g^+_\ell \hnem \hbar^\ell
\nem\hnem \left(
\frac{-2\mem\three{\mem\cdot\mem}a}{\vphantom{\adjustbox{raise=0.2em}{$\bpsi$}}\bpsi\psi}
\right)^{\kern-0.17em\ell}
\left(\hnem
1 - \frac{1}{(\ell-1)!}\mem \Gamma\big(\ell\hem\big|\hem\bpsi\psi + \three{\mem\cdot\mem}a\big)
\nem\kern-0.015em\right)
\nem\nem\right)
\,.
\end{split}
\end{align}

We have plugged in (<ref>) to (<ref>)
and then placed $\mathe^{\three\cdot a}$ in front of it by hand
while omitting $e_h\hem m_0 x^h\mem$ to avoid clutter.
In the large $\bpsi\psi$ limit, this agrees with the last line of (<ref>)
so that (<ref>)-(<ref>) are reproduced.

It is not difficult to

why this trick works
at the level of computation.
For a deeper reason,
might be plausible

to relate it to the fact that
the quantum amplitude of the minimally coupled particle, derived in (<ref>),
is supported on $\one_\a{}^I = \two_\a{}^I$.

§ HAMILTONIAN PERTURBATION THEORY

Finally, we would like to comment that
it is also possible to develop the textbook perturbation theory
(which perturbs the Hamiltonian, not the symplectic structure)
for the massive twistor particle.
The purpose of this digression
is to complement our symplectic perturbation

and point out possibilities of future applications,
e.g., applying the KMO'C formalism [13].
We only
the simplest example:
scattering of a minimally coupled
particle in the plus-helicity electromagnetic plane wave.

Canonical quantization

We first-quantize the massive twistor particle
in the reparameterization gauge (<ref>)
where the Hamiltonian is $H = \minime (\bDelta {\mem+\mem} \Delta) + m_0$
and treat it as a standard quantum-mechanical system
with “absolute time.”

In order to perform canonical quantization,
one should figure out what the canonical coordinates are.
A holomorphic electromagnetic field $A_{\a\da}\hnem(z)\mem dz^{\da\a}$
translates to $-i\hem
A_{\smash{\a\dc}\vphantom{\da}}\hnem(z)\mem \rho^{\dc\rmA}\hem dZ_\rmA{}^I$
in the notation (<ref>).
Taking $i\mem \bZ_I{}^\rmA dZ_\rmA{}^I$ for the free-theory symplectic potential,
we find that the “kinetic” and “canonical” twistor variables are related as

\begin{align}
\label{eq:kinZcanZ}
\mem=\mem (Z^{\text{can}})_\rmA{}^I
\,,\quad
\mem=\mem (\bZ^{\text{can}})_I{}^\rmA
\mem+\mem
\mem A_{\smash{\a\dc}\vphantom{\da}}\hnem(z^{\text{kin}})\,
\,.
\end{align}

From now, we exclusively work with the canonical variables
(unlike in the main text)
and drop the “can” subscript to avoid clutter.
Then (<ref>) boils down to

\begin{align}
\begin{split}
\lambda_\a{}^I
&= (\lambda^\text{kin})_\a{}^I
\,,\quad
\mu^{\da I}
= (\mu^\text{kin})^{\da I}
\,,\\
\rambda_{I\da}
&= (\rambda^\text{kin})_{I\da}
- q(\lambda^{-1})_I{}^\b A_{\smash{\b\da}\vphantom{\da}}\hnem(z)
\,,\\
\bmu_I{}^\a
&= (\bmu^\text{kin})_I{}^\a
- q(\lambda^{-1})_I{}^\b A_{\smash{\b\da}\vphantom{\da}}\hnem(z)\mem
% \,,
\,.
\end{split}
\end{align}

While $z^{\da\a} {\mem=\mem} (\mu\lambda^{-1})^{\da\a} {\mem=\mem} (z^{\text{kin}})^{\da\a}$,
$\bz^{\da\a} {\mem=\mem} (\rambda^{-1}\hhnem\bmu)^{\da\a}$ differs from $(\bz^{\text{kin}})^{\da\a}$
due to coupling to the holomorphic gauge field.

The Hamiltonian
$H = \minime (\det(\rambda^\text{kin}) {\mem+\mem} \det(\lambda^\text{kin}))$
gets perturbed in this canonical setting as

H = H^\circ + H'
%     \,,
% \end{align}
where $H^\circ = \minime (\det(\rambda) + \det(\lambda)) + m_0$ and
$H' = H'_1 + H'_2$:

\begin{align}
% H' &= H'_1 + H'_2
% \,,\quad
H'_1 :=
\frac{\hem\bDelta\hem}{2}\mem
qA_{\smash{\b\db}\vphantom{\da}}\hnem(z)\mem (p{}^{-1})^{\db\b}
\,,\quad
H'_2 :=
-\frac{1}{4\Delta}\mem q^2\mem \e^{\a\b} A_{\a\da}\hnem(z)\mem A_{\smash{\b\db}\vphantom{\da}}\hnem(z)\mem \be^{\da\db}
\,.
\end{align}

We restrict our attention to the leading order
and discard $H'_2$.
For the self-dual plane wave (<ref>), the $\mathcal{O}(q^1)$ interaction Hamiltonian becomes

\begin{align}
\label{eq:H'1}
= \frac{q}{\sqrt{2} \Delta}\mem
% \i^\a p_{\a\da} \bo^\da
\langle\i|p|\bo\rsq
% \,\mathe^{-i\bozo}
\,\mathe^{-i\mem \bo \mu\lambda^{-1}\hnem\o}
\,.
\end{align}

There are many kinds of eigenstates
in the free theory
that we can consider:
coherent states $\Bra{\bZ}$ and $\Ket{Z}$,
frame eigenstates $\Ket{\lambda\rambda}$,
and simultaneous eigenstates of $\hat{p}_{\a\da}$ and $\hat{\vec{W}}{}^2$,
which might be called “Wigner states.”
For our scattering problem, we identify $\Ket{\lambda\rambda}$ as the relevant in-state.
Using the convention
$\{Z_\rmA{}^I,\bZ_J{}^\rmB\} = (1/\ihbar)\mem \delta_\rmA{}^\rmB \delta_J{}^I$
$\smash{[\hat{Z}_\rmA{}^I,\hat{\bZ}_J{}^\rmB]} = \delta_\rmA{}^\rmB \delta_J{}^I$
(see footnote <ref>),
the canonically quantized operators are realized as

\begin{align}
\begin{split}
\Bra{\lambda\rambda}\hem \hat{\lambda}_\a{}^I
= \lambda_\a{}^I \Bra{\lambda\rambda}
\Bra{\lambda\rambda}\hem \hat{\rambda}_{I\da}
= \rambda_{I\da} \Bra{\lambda\rambda}
\,,\\
\Bra{\lambda\rambda}\hem \hat{\mu}^{\da I}
= -i\mem
\frac{\partial}{\partial\rambda_{I\da}}
\Bra{\lambda\rambda}
\Bra{\lambda\rambda}\hem \hat{\bmu}_I{}^\a
= -i\mem
\frac{\partial}{\partial\lambda_\a{}^I}
\Bra{\lambda\rambda}
\,.
\end{split}
\end{align}

AHH amplitude from first quantization
From the interaction Hamiltonian (<ref>),
the matrix element

is given by

\begin{align}
\label{eq:can-amp.1}
\Bra{\lambda'\rambda{}'} \hat{T} \Ket{\lambda\rambda}
-\nem\int d\t\,\,
\Bra{\lambda'\rambda{}'}
\mem
\mathe^{-(\t/\ihbar)\hat{H}^\circ}
{\hat{H}'_1 \,}
\mathe^{(\t/\ihbar)\hat{H}^\circ}
\Ket{\lambda\rambda}
+ \mathcal{O}(q^2)
\,,\\
% 2\pi\hbar\mem\hem
\deltabar(E'{\hhem-\mem}E)
\bigg[\hhem\mem{
\langle\i|p|\bo\rsq\mem
\exp\nem\nem\bigg({
\three_{\a\da}\mem
\frac{\partial}{\partial\rambda{}'_{I\da}}
\BraKet{\lambda'\rambda{}'}{\lambda\rambda}
+ \mathcal{O}(q^2)
\,,
\nonumber
\end{align}

where $E := \minime(\det(\rambda)+\det(\lambda))$,
$E' := \minime(\det(\rambda{}')+\det(\lambda'))$,
and $\deltabar(\xi) := 2\pi\hbar\mem\hem \delta(\xi)$.
The exponential operator implements a translation of the right-handed spin frame:

\begin{align}
\begin{split}
\label{eq:can-amp.2}
\exp\nem\nem\bigg({
\three_{\a\da}\mem
\frac{\partial}{\partial\rambda{}'_{I\da}}
\BraKet{\lambda'\rambda{}'}{\lambda\rambda}
\BraKet{\lambda', \rambda{}' {+} \lambda^{-1}\three }{\lambda\rambda}
\,,\\
\delta^{(4)}\hnem(\lambda' {\hem-\mem} \lambda)\,
\delta^{(4)}\hnem(\mem{
\rambda{}' + \lambda^{-1}\three - \rambda
\,.
\end{split}
\end{align}

When combined with $\deltabar(E'{\hhem-\mem}E)$,
this product of delta functions
precisely describes the heavenly kinematics (<ref>).[
When dealing with delta functions of the complexified spinor-helicity variables and their determinants, we are implicitly working in the $(2,2)$ signature.
Especially, one finds that $E' {\hhem-\mem} E \hem=\hem \three{\mem\cdot\mem}p\mem/\Delta$.

Then we obtain

\begin{align}
\label{eq:can-amp.3}
\Bra{\lambda'\rambda{}'} \hat{T} \Ket{\lambda\rambda}
% \,=\,
% -\Delta\,
% \deltabar(\three{\mem\cdot\mem}p)\,
% \delta^{(4)}\hnem(\lambda' {\hem-\mem} \lambda)\,
% \delta^{(4)}\hnem\big(\mem{
%     \lsq\rambda|
%     + \lsq\rambda \bthree\rsq \lsq\bthree| / mx
%     - \lsq\rambda{}'|
% }\mem\big)
% \,
% \frac{q}{\sqrt{2}\Delta}\mem
% \langle\i|p|\bo\rsq\mem
% + \mathcal{O}(q^2)
\sqrt{2}q\hem |\Delta|x\,\mem
\deltabar(2\mem\three{\mem\cdot\mem}p)\,
\delta^{(4)}\hnem(\lambda {\hem-\mem} \lambda')\,
% \delta^{(4)}\hnem\big(\mem{
%     \lsq\rambda|
%     {\mem+\mem} \lsq\rambda \bthree\rsq \lsq\bthree| / |\Delta|x
%     {\mem-\mem} \lsq\rambda{}'|
% }\mem\big)
\delta^{(4)}\hnem\bigg({
\lsq\rambda|
\lsq\rambda|
\frac{|\bthree\rsq\lsq\bthree|}{|\Delta|x}
{\mem-\mem} \lsq\rambda{}'|
+ \mathcal{O}(q^2)
% \,,
\,.
\end{align}

Observe how the first-quantized massive twistor operator algebra
in (<ref>)-(<ref>)
elegantly realizes Guevara, Ochirov, and Vines [8]'s exponential operator.
We can take (<ref>) as a straight-up derivation
of Arkani-Hamed, Huang, and Huang [3]'s
minimally coupled
on-shell amplitude
from the first-quantized theory.

To be precise, it is actually not yet on-shell,

the $x$-factor could be well-defined
in virtue of the delta function $\deltabar(2\mem\three{\mem\cdot\mem}p) = \deltabar(E'{\hhem-\mem}E) / \Delta$.
If we
constrain the in- and out-states
to lie on

the mass shell,

\begin{align}
(\hat{\bDelta} {\mem-\mem} m_0) \Ket{\one\bone}
= 0
\,,\quad
(\hat{\Delta} {\mem-\mem} m_0) \Ket{\one\bone}
= 0
% \,,\quad
% \hat{\lambda}_\a{}^I \Ket{\one\bone}
% = \one_\a{}^I \Ket{\one\bone}
% \,,\quad
% \hat{\rambda}_{I\da} \Ket{\one\bone}
% = \bone_{I\da} \Ket{\one\bone}
\,,
\end{align}

we can

the matrix element with [3]'s amplitude:

\begin{align}
\Bra{\two\btwo} \hat{T} \Ket{\one\bone}
\sqrt{2}q\hem m_0x\,\mem
\deltabar(2\mem\three{\mem\cdot\mem}\one)\,
\delta^{(4)}\hnem\big(\mem{
|\one\rangle {\mem-\mem} |\two\rangle
\delta^{(4)}\hnem\bigg({
\lsq\bone|
\lsq\bone|
\frac{|\bthree\rsq\lsq\bthree|}{m_0x}
{\mem-\mem} \lsq\btwo|
+ \mathcal{O}(q^2)
% \,,
\,.
\end{align}

We emphasize again that the amplitude is supported on the heavenly kinematics
so that it is indeed tempting to introduce the
diagrammatic notations

Perhaps a certain realization of on-shell recursion in the massive twistor space
could take the heavenly kinematics as basic building blocks.

Going further, we can try to reproduce the impulses in section <ref>
by applying the KMO'C formalism [13]
in this first-quantized setting.
Yet another direction
is non-minimal couplings.
Even though the relation between “kinematic” and “canonical” variables becomes unclear,
we could
work with higher-spin interaction Hamiltonians
constructed from real-spacetime fields.
We leave these topics for future research.

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