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?


1 Answer 1


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).

But while scalar and gauge fields end up each with their own components of the boundary they do comprise a single interacting field theory. In particular, gauge transformations that are behind these asymptotic symmetries act in the bulk on both fields simultaneously. So it is natural that the expressions for both asymptotic regions would involve both fields.


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