# How are the authors obtaining the asymptotic form of the sympletic form for the Maxwell + massive field system?

I've been studying the paper "Asymptotic symmetries of QED and Weinberg’s soft photon theorem" by Campiglia & Laddha and there is one step in their analysis I'm being unable to understand. I shall quote the relevant discussion:

We now describe the sympletic structure of the Maxwell + massive field system. The covariant phase space sympletic density reads $$\omega^\mu(\delta,\delta')=\sqrt{g}\left(\delta {\cal F}^{\mu\nu}\delta'{\cal A}_\nu + (D^\mu\delta\varphi)^\ast \delta '\varphi + \text{c.c}\right) - \delta\leftrightarrow\delta'.\tag{25}$$ Given a solution to the field equation $$({\cal A}_\mu,\varphi)$$ and variations $$\delta,\delta'$$ thereof, we want to evaluate the sympletic product in terms of the asymptotic fields by: $$\Omega(\delta,\delta')\equiv \lim_{t\to+\infty}\int_{\Sigma_t} dS_\mu \omega^\mu(\delta,\delta'),\tag{26}$$ with $$\Sigma_t$$ a $$t = \text{constant}$$ Minkowski time slice. The asymptotic form of $$\omega^t$$ depends on how the $$t=\text{constant}$$ fields are parameterized in the radial direction as $$t\to \infty$$. If one keeps $$u = t-r$$ constant, conditions (18) and (19) imply: \begin{align}\omega^t &= \omega^r+\omega^u,\\ \omega^r &= \sqrt{\gamma}\gamma^{MN}\partial_u \delta' A_N\delta A_M-\delta\leftrightarrow\delta' + O(t^{-1})\tag{27}\\ \omega^u &= O(t^{-1}).\end{align} If on the other hand one keeps $$r/t$$ constant, conditions (22) and (24) imply that $$\omega^t$$ coincides with the free massive field sympletic density (12) up to terms that vanish in the $$t\to\infty$$ limit. We thus conclude that: $$\Omega(\delta,\delta')=\Omega_A(\delta,\delta')+\Omega_\phi(\delta,\delta'),\tag{28}$$ where $$\Omega_A(\delta,\delta')=\int_{\mathcal{I}^+}\sqrt{\gamma} du(\delta_A\partial_u \delta' A^A - \delta\leftrightarrow \delta')\tag{29}$$ is the standard sympletic product of the Maxwell field radiative phase space $$\Gamma^A$$ and $$\Omega_\phi(\delta,\delta')$$ the free massive field sympletic product as given in the RHS of Eq. (13).

Now I'm failing to understand this idea of "first taking $$u=t-r$$ constant and then $$r/t$$ constant" in order to get the two terms in Eq. (28).

In my mind, you either take $$u=t-r$$ constant, which would lead one to conclude that $$\Omega(\delta,\delta')=\Omega_A(\delta,\delta')$$, or take $$r/t$$ constant, which would lead, one to conclude that $$\Omega(\delta,\delta')=\Omega_\phi(\delta,\delta')$$. This is obviously wrong, so I'm missing something.

One obvious idea seems to be: well, split the integral in (26) in a sum of two: one containing the $${\cal A}_\mu$$ part and the other containing the $$\varphi$$ part. But I think that's not what the authors mean. In fact, the $${\cal A}_\mu$$ part would not involve $$\varphi$$ and yet the authors claim the behavior of $$\varphi$$ near $${\cal I}^+$$ (conditions (19)) has been used to get the first $$\Omega_A$$ part.

So what is going on? What am I missing? How does one get Eq. (28) following the authors procedure?

The theory being considered here is electrodynamics with a massive scalar field. If we consider the “distant future” asymptotic state of any sort of interaction then there would be two types of particles: quanta of scalar field moving with constant velocities (less than $$c$$), and photons.
Scalar field quanta end up at timelike future infinity $$\mathcal{i}^{+}$$. This is what the limit $$r/t=\mathrm{const}$$ catches (with the obvious interpretation of that constant as the velocity modulus). Note, that in the conformal treatments of infinities à la Penrose timelike infinity of Minkowski space ends up shrunken to a point, but here we have a manifold description of this infinity with the natural coordinates being velocity components $$\vec{v}$$.
Photons in the future end up at future null infinity $$\mathscr{I}^{+}$$. This corresponds to the limit $$t-r=\mathrm{const}$$ and this is the part that is most emphasized in conformal treatments of asymptotically flat spacetimes, and is a particular focus of much of recent activities regarding infrared structures of field theories (and so we would not be discussing it further here).