# Fermionic asymptotic symmetries in massless QED

Adrián Agriela<EMAIL_ADDRESS>Instituto de Física, Facultad de
Ciencias, Universidad de la República
Iguá 4225, Montevideo, Uruguay  Instituto de Física, Facultad de Ingeniería,
Universidad de la República
Julio Herrera y Reissig 565, Montevideo, Uruguay  Miguel Campiglia
<EMAIL_ADDRESS>Instituto de Física, Facultad de Ciencias, Universidad de
la República
Iguá 4225, Montevideo, Uruguay

###### Abstract

We consider soft electrons in massless QED at tree-level. The emission
amplitude at leading order in the soft electron energy factorizes in a way
similar to the soft photon case. We recast the soft electron factorization
formula as a Ward identity of an asymptotic charge. This leads to the first
example of an asymptotic fermionic symmetry in a theory with no conventional
supersymmetry, suggesting that tree-level massless QED may posses an
asymptotic supersymmetry algebra. Although our approach does not yet allow us
to completely characterize the algebra, it suggests that subleading soft
photons should feature in the anticommutator of two fermionic symmetry
generators.

###### Contents

  1. I Introduction
  2. II Preliminaries
    1. II.1 Conventions
    2. II.2 Asymptotic fields at null infinity
      1. II.2.1 Outgoing null coordinates
      2. II.2.2 Fall-offs in $r$
    3. II.3 Radiative phase space
      1. II.3.1 Fall-offs in $u$
      2. II.3.2 Radiative PBs
    4. II.4 Asymptotic Fock space
    5. II.5 Bosonic (asymptotic) symmetries
      1. II.5.1 Lorentz
      2. II.5.2 Translations
      3. II.5.3 Large $U(1)$ gauge
      4. II.5.4 Axial rotations
  3. III Soft electrons and fermionic asymptotic charges
  4. IV On fermionic asymptotic symmetries
    1. IV.1 Symmetry action from radiative PBs
    2. IV.2 Algebra relations with bosonic symmetries
      1. IV.2.1 Lorentz
      2. IV.2.2 Translations
      3. IV.2.3 Large $U(1)$ gauge
      4. IV.2.4 Axial rotations
  5. V Outlook
  6. A Asymptotic vs. momentum-space Fock operators
  7. B Soft photons and soft electrons
    1. B.1 Soft photon
    2. B.2 Soft electron
  8. C Naive commutator of fermionic charges and subleading soft photons
    1. C.1 Asymptotic symmetries for subleading soft photon
    2. C.2 Naive commutator of two fermionic generators

##  I Introduction

Asymptotic states of massless particles have their natural home at null
infinity penrose . At this boundary of spacetime it is possible to unveil
symmetries that are otherwise obscure from a bulk perspective asymquant ;
stromlectures . As first shown by Strominger and collaborators stromST ;
stromVir ; stromQED , these asymptotic symmetries manifest themselves in
scattering amplitudes via so-called soft theorems. Conversely, soft theorems
in scattering amplitudes can often be interpreted as arising from asymptotic
symmetries, see e.g. stromLow ; stromfermionic ; lysovgravitino ;
averygravitino ; subsub . In this article we present what may be the simplest
example of an asymptotic _fermionic_ symmetry. We consider tree-level massless
Quantum Electrodynamics (QED), in which there is a notion of soft electrons,
i.e. electrons with vanishingly small energy.111Alternatively, we are studying
massive electrons in a regime where the soft energy $E_{\textrm{soft}}$ is
much larger than the electron mass yet much smaller than all other energies
involved in the process: $m_{e}\ll E_{\textrm{soft}}\ll E_{\textrm{hard}}$.
The emission amplitude for such soft electrons is given to leading order by a
soft electron theorem. We will recast such theorem as a Ward identity of
fermionic asymptotic charges, thus suggesting the existence of asymptotic
fermionic symmetries. At this stage, however, we are unable to fully
characterize the underlying symmetry algebra. In particular, we lack a
trustworthy evaluation of the commutator between fermionic symmetries. We will
describe what the obstacles are and discuss possible strategies to overcome
them.

Fermionic asymptotic symmetries have already been discussed in other contexts.
They naturally arise in supergravity theories Awada:1985by ; Henneaux:2020ekh
; Fuentealba:2020aax ; Fuentealba:2021xhn ; banerjee  where they are
associated to soft gravitinos lysovgravitino ; averygravitino . In
stromfermionic , Dumitrescu, He, Mitra and Strominger identified asymptotic
fermionic charges in supersymmetric abelian gauge theories. Our work follows
closely their analysis, with the electron field here playing the role of
photino field there. A comprehensive discussion of (conformally) soft fermions
can be found in Pano:2021ewd .

The organization of the paper is as follows. In the next section we introduce
notation and review basic concepts on radiative phase spaces and asymptotic
symmetries, in the context of massless QED. In section III we present a soft
electron theorem and interpret it as a Ward identify of an asymptotic
fermionic charge. In section IV we discuss various aspects of the associated
fermionic symmetries, including the commutation relations with bosonic
symmetries. We conclude in section V, where we highlight the open questions
left for future work. Additional material is given in three appendices: In
appendix A we review the relationship between the momentum space and null
infinity descriptions of fields. In appendix B we review the soft photon
theorem and derive the analogous soft electron theorem. In appendix C we
present a preliminary exploration on the non-linear structure of the fermionic
symmetry, and observe how a naive evaluation of the commutator of two
fermionic symmetries displays similarities with the asymptotic symmetry
associated to subleading soft photons.

##  II Preliminaries

###  II.1 Conventions

The elementary field variables for QED are the $U(1)$ gauge field
$\mathcal{A}_{\mu}$ and the anticommuting Dirac spinor $\Psi$. In the massless
case the lagrangian density is222We follow conventions from peskin  modulo a
sign in the definition of the coupling constant $e$.

$\mathcal{L}=\sqrt{-\eta}\left(-\frac{1}{4}\mathcal{F}^{\mu\nu}\mathcal{F}_{\mu\nu}+i\overline{\Psi}\gamma^{\mu}\mathcal{D}_{\mu}\Psi\right),$
(1)

where $\sqrt{-\eta}$ is the Minkowski volume element,
$\mathcal{F}_{\mu\nu}=\partial_{\mu}\mathcal{A}_{\nu}-\partial_{\nu}\mathcal{A}_{\mu}$
is the field strength, $\gamma^{\mu}$ Dirac matrices,
$\overline{\Psi}=\Psi^{\dagger}\gamma^{0}$ and

$\mathcal{D}_{\nu}=\partial_{\mu}+ie\mathcal{A}_{\mu}$ (2)

the gauge covariant derivative. The electric current is defined as

$\mathcal{J}^{\mu}=e\overline{\Psi}\gamma^{\mu}\Psi.$ (3)

Taking variations of the lagrangian density, one finds

$\delta\mathcal{L}=eom+\partial_{\mu}\theta^{\mu}(\delta)$ (4)

where

$eom=(\partial_{\mu}\mathcal{F}^{\mu\nu}-\mathcal{J}^{\nu})\delta\mathcal{A}_{\nu}+(i\delta\overline{\Psi}\gamma^{\mu}\mathcal{D}_{\mu}\Psi+c.c.)$
(5)

yield the field equations and

$\theta^{\mu}(\delta)=\sqrt{-\eta}(-\mathcal{F}^{\mu\nu}\delta\mathcal{A}_{\nu}+i\overline{\Psi}\gamma^{\mu}\delta\Psi)$
(6)

is the symplectic potential current.

Besides Poincare (and in fact Conformal) symmetries, the theory is invariant
under local gauge transformations

$\delta_{\Lambda}\mathcal{A}_{\mu}=\partial_{\mu}\Lambda,\quad\delta_{\Lambda}\Psi=-ie\Lambda\Psi,$
(7)

as well as global axial rotations

$\delta_{\text{A}}\mathcal{A}_{\mu}=0,\quad\delta_{\text{A}}\Psi=i\gamma^{5}\Psi.$
(8)

The latter famously displays a 1-loop anomaly abj , but for the purposes of
our tree-level discussion, we will regard (8) as an exact symmetry.

###  II.2 Asymptotic fields at null infinity

####  II.2.1 Outgoing null coordinates

In order to describe the fields near (future) null infinity, it is convenient
to work in outgoing null coordinates. These can be defined as follows. First,
assign a future null direction $q^{\mu}$ to every point $x\equiv(z,\bar{z})$
on the celestial sphere,

$q^{\mu}(x):=\frac{1}{\sqrt{2}}\left(1+|z|^{2},z+\bar{z},-i(z-\bar{z}),1-|z|^{2}\right),$
(9)

The specific choice (9) leads to a flat conformal frame on the celestial
sphere, see e.g. Kapec:2017gsg  for a discussion on other possible frames.
Next, choose a reference null vector $k^{\mu}$ transverse to $q^{\mu}$ that
specifies the “flow of time”,

$k^{\mu}=\frac{1}{\sqrt{2}}\left(1,0,0,-1\right),\quad k^{\mu}q_{\mu}(x)=-1.$
(10)

Finally, parametrize cartesian coordinates $X^{\mu}$ by $(r,u,x)$ according to

$X^{\mu}(r,u,x)=rq^{\mu}(x)+uk^{\mu},$ (11)

in terms of which the spacetime metric takes the form

$dX^{\mu}dX_{\mu}=-2dudr+2r^{2}dzd\bar{z}.$ (12)

Future null infinity $\mathcal{I}$ is reached by taking $r\to\infty$ with
constant $(u,x)$. One can similarly define retarded null coordinates that are
adapted to past null infinity. We will however focus our discussion on fields
at future null infinity, with the understanding that a parallel construction
is available at past null infinity.

####  II.2.2 Fall-offs in $r$

Near null infinity the electromagnetic field is described by the leading
transversal components asymquant ; stromlectures ,

$\mathcal{A}_{z}(r,u,x)\stackrel{{\scriptstyle
r\to\infty}}{{=}}A_{z}(u,x)+\cdots,$ (13)

where $A_{z}$ is regarded as a gauge field on $\mathcal{I}$. The massless
Dirac field fall-offs have been discussed in stromfermionic . As for other
massless fields, it decays as the inverse power of the radial coordinate.333We
are assuming free-field fall-offs, which suffice for the tree-level
considerations of this work. Loop corrections may imply slower fall-offs, see
e.g. sayali . The Dirac equation imposes restrictions on the leading spinor
components, leaving only two independent asymptotic fields:444Further details
are given in the discussion following Eq. (130).

$\Psi\stackrel{{\scriptstyle
r\to\infty}}{{=}}\frac{1}{2^{1/4}r}\begin{pmatrix}-\bar{z}\psi_{-}\\\
\psi_{-}\\\ \psi_{+}\\\ z\psi_{+}\end{pmatrix}+\cdots,$ (14)

where the overall normalization is chosen for later convenience and
$\psi_{\pm}$ are regarded as complex fermionic fields on $\mathcal{I}$. Upon
quantization, they describe electrons of positive/negative helicity.
Similarly, $A_{z}$/$A_{\bar{z}}$ describe photons of positive/negative
helicity. We shall use the notation

$A_{+}:=A_{z},\quad A_{-}:=A_{\bar{z}}.$ (15)

Notice that, unlike $\psi_{\pm}$, the gauge field satisfies the reality
condition $A^{*}_{+}=A_{-}$.

###  II.3 Radiative phase space

Given the fall-offs from the previous section, we can evaluate the symplectic
potential current (6) at null infinity. For the relevant component $\mu=r$ one
finds

$\lim_{r\to\infty}\theta^{r}(\delta)=\dot{A}_{z}\delta
A_{\bar{z}}+\dot{A}_{\bar{z}}\delta
A_{z}+i\bar{\psi}_{+}\delta\psi_{+}+i\bar{\psi}_{-}\delta\psi_{-}$ (16)

where $\dot{A}_{z}\equiv\partial_{u}A_{z}$ and $\bar{\psi}_{\pm}$ is the
complex conjugate of $\psi_{\pm}$. Taking a second variation in (16) and
integrating over $(u,z,\bar{z})$ we obtain the symplectic structure at
$\mathcal{I}$ AS ,

$\Omega:=\sum_{s=\pm}\int_{\mathcal{I}}dud^{2}x\left(\delta\dot{A}_{-s}\wedge\delta
A_{s}+i\delta\bar{\psi}_{s}\wedge\delta\psi_{s}\right),$ (17)

where we used the notation (15) for the asymptotic gauge field. Evaluating
(17) on two variations $\delta_{1}$ and $\delta_{2}$ one has

$\Omega(\delta_{1},\delta_{2})=\sum_{s}\int_{\mathcal{I}}\left(\delta_{1}\dot{A}_{-s}\delta_{2}A_{s}+i\delta_{1}\bar{\psi}_{s}\delta_{2}\psi_{s}\right)-(\delta_{1}\leftrightarrow\delta_{2}).$
(18)

In the above expressions it is important that $\psi_{\pm}$ are regarded as
anticommuting fields. In particular, the reality of the symplectic form
follows from the property
$\overline{\psi_{1}\psi_{2}}=\bar{\psi}_{2}\bar{\psi}_{1}$.

####  II.3.1 Fall-offs in $u$

In order for (17) to be well-defined, we must impose fall-offs conditions on
the fields as $|u|\to\infty$. We shall henceforth assume them to be

$\displaystyle A_{s}(u,x)$
$\displaystyle\stackrel{{\scriptstyle|u|\to\infty}}{{=}}$ $\displaystyle
O(1)+O(1/|u|^{\epsilon})$ (19) $\displaystyle\psi_{s}(u,x)$
$\displaystyle\stackrel{{\scriptstyle|u|\to\infty}}{{=}}$ $\displaystyle
O(1/|u|^{1+\epsilon})$ (20)

for some $\epsilon>0$. Condition (19) is slightly more general than the
typical scattering fall-off, which corresponds to $\epsilon=1$ laddhasen .
Condition (20) is stronger than a minimal requirement for convergence of (17)
(for which it would be enough a fall-off faster than $1/|u|^{1/2}$). We
require (20) to ensure finiteness of the asymptotic fermionic charge defined
in section III.

####  II.3.2 Radiative PBs

From the symplectic structure (17) one can obtain the elementary non-trivial
Poisson brackets (PBs)555Our conventions are as follows. The PBs between two
functions $F$ and $G$ is given by
$\\{F,G\\}:=X_{G}(F)=-X_{F}(G)=\Omega(X_{G},X_{F})$ where $X_{F}$ is defined
by the condition $\Omega(\delta,X_{F})=\delta F$ (and similarly for $X_{G}$).
If $F$ and $G$ are fermionic, there are additional signs that can be
determined by requiring the grasmannian Leibinitz rule on PBs, namely
$\\{F_{1}F_{2},G\\}=F_{1}\\{F_{2},G\\}\pm\\{F_{1},G\\}F_{2}$ where the minus
occurs if both $F_{2}$ and $G$ are fermionic. In particular the PBs between
two fermionic functions is symmetric rather than antisymmetric. We refer to
chapter 6 of HT  for further details on fermionic PBs.

$\displaystyle\\{A_{s}(u,x),\dot{A}_{-s}(u^{\prime},x^{\prime})\\}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\delta(u-u^{\prime})\delta^{(2)}(x,x^{\prime}),$ (21)
$\displaystyle\\{\psi_{s}(u,x),\bar{\psi}_{s}(u^{\prime},x^{\prime})\\}$
$\displaystyle=$
$\displaystyle-i\delta(u-u^{\prime})\delta^{(2)}(x,x^{\prime}).$

We recall however a well known subtlety with the gauge field PBs stromQED ,
which is the occurrence of a $1/2$ discontinuity that can be expressed as

$\int^{\infty}_{-\infty}du\\{\cdot,\dot{A}_{s}(u,x)\\}=\frac{1}{2}\\{\cdot,\int^{\infty}_{-\infty}du\dot{A}_{s}(u,x)\\},$
(22)

where $\\{\cdot,F\\}\equiv X_{F}$ is the Hamiltonian vector field of a
functional $F$ (see footnote 5). A fix to this problem was proposed in
stromQED  via the isolation of the zero mode component of the gauge field and
the introduction of boundary terms in the symplectic structure. For simplicity
we will continue to work with the standard radiative phase space symplectic
structure while keeping care when needed of the aforementioned subtlety.

###  II.4 Asymptotic Fock space

To obtain the asymptotic Fock space of photons and massless electrons, we
start by considering the Fourier transform of the fields with respect to the
$u$-variable,

$\displaystyle
A_{s}(u,x)=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\tilde{A}_{s}(\omega,x)e^{-i\omega
u},$ (23)
$\displaystyle\psi_{s}(u,x)=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\tilde{\psi}_{s}(\omega,x)e^{-i\omega
u},$
$\displaystyle\bar{\psi}_{s}(u,x)=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\tilde{\bar{\psi}}_{s}(\omega,x)e^{i\omega
u}.$

The brackets (21) imply

$\displaystyle\\{\tilde{A}_{s}(\omega,x),\tilde{A}_{-s}(\omega^{\prime},x^{\prime})\\}$
$\displaystyle=$
$\displaystyle\frac{i\pi}{\omega^{\prime}}\delta(\omega+\omega^{\prime})\delta^{(2)}(x,x^{\prime}),$
(24)
$\displaystyle\\{\tilde{\psi}_{s}(\omega,x),\tilde{\bar{\psi}}_{s}(\omega^{\prime},x^{\prime})\\}$
$\displaystyle=$
$\displaystyle-i2\pi\delta(\omega-\omega^{\prime})\delta^{(2)}(x,x^{\prime}).$

Notice that the information of $A_{s}(u,x)$ is contained in
$\tilde{A}_{s}(\omega,x),\omega>0$ since
$\tilde{A}_{s}(-\omega,x)=(\tilde{A}_{-s}(\omega,x))^{*}$ due to the reality
of the gauge field. On the contrary, there is no relation between the positive
and negative frequency components of $\tilde{\psi}_{s}(\omega,x)$. An
independent set of mode functions is then given by:

$\displaystyle a_{s}(\omega,x)$ $\displaystyle:=$
$\displaystyle\tilde{A}_{s}(\omega,x)$ $\displaystyle,\quad\omega>0$ (25)
$\displaystyle b_{s}(\omega,x)$ $\displaystyle:=$
$\displaystyle\tilde{\psi}_{s}(\omega,x)$ $\displaystyle,\quad\omega>0$
$\displaystyle c_{s}(\omega,x)$ $\displaystyle:=$
$\displaystyle\tilde{\bar{\psi}}_{-s}(-\omega,x)$
$\displaystyle,\quad\omega>0.$

Upon quantization, these become the annihilation Fock operators of photons,
electrons and positrons respectively.666The normalization however is different
from the standard momentum-space Fock operators, see appendix A. The (anti)
commutation relations of these operators with their hermitian adjoints are
dictated by $i$ times the PBs (24), from which one finds

$\displaystyle{[a_{s}(\omega,x),a^{\dagger}_{s}(\omega^{\prime},x^{\prime})]}$
$\displaystyle=\frac{\pi}{\omega}\delta(\omega-\omega^{\prime})\delta^{(2)}(x,x^{\prime}),$
(26)
$\displaystyle[b_{s}(\omega,x),b^{\dagger}_{s}(\omega^{\prime},x^{\prime})]$
$\displaystyle=[c_{s}(\omega,x),c^{\dagger}_{s}(\omega^{\prime},x^{\prime})]=2\pi\delta(\omega-\omega^{\prime})\delta^{(2)}(x,x^{\prime}).$

It will be useful for later purposes to define angular-density operators for
each kind of particle:

$\displaystyle\rho^{A}_{s}(x)$ $\displaystyle:=$
$\displaystyle\int_{0}^{\infty}\frac{d\omega}{\pi}\omega
a^{\dagger}_{s}(\omega,x)a_{s}(\omega,x)$ (27)
$\displaystyle\rho^{\psi}_{s}(x)$ $\displaystyle:=$
$\displaystyle\int_{0}^{\infty}\frac{d\omega}{2\pi}b^{\dagger}_{s}(\omega,x)b_{s}(\omega,x)$
(28) $\displaystyle\rho^{\bar{\psi}}_{s}(x)$ $\displaystyle:=$
$\displaystyle\int_{0}^{\infty}\frac{d\omega}{2\pi}c^{\dagger}_{s}(\omega,x)c_{s}(\omega,x)$
(29)

as well as particle-number operators

$N^{X}_{s}:=\int d^{2}x\rho^{X}_{s}(x),\quad X=\\{A,\psi,\bar{\psi}\\}.$ (30)

We finally note the identity

$\int_{-\infty}^{\infty}du\bar{\psi}_{s}(u,x)\psi_{s}(u,x)=\rho^{\psi}_{s}(x)-\rho^{\bar{\psi}}_{-s}(x).$
(31)

###  II.5 Bosonic (asymptotic) symmetries

In this section we review bosonic symmetries of tree-level massless QED at
null infinity: Poincare, large $U(1)$ gauge and axial rotations. We leave out
of the discussion other bosonic symmetries that are more challenging to
describe at null infinity: Those arising from subn-leading soft photons777See
however appendix C for a preliminary incorporation of subleading soft photon
symmetries. hamada ; Li:2018gnc  as well as 4-dimensional conformal symmetry.

####  II.5.1 Lorentz

Infinitesimal Lorentz transformations (or more generally superrotations) near
null infinity are parametrized by holomorphic vector fields $Y(z)\partial_{z}$
(and their complex conjugate) by

$\xi_{Y}=Y(z)\partial_{z}+\frac{1}{2}Y^{\prime}(z)(u\partial_{u}-r\partial_{r})+\cdots,$
(32)

They act on fields via standard Lie derivatives, which for spinors include an
internal rotation (see e.g. stromfermionic )

$\displaystyle\delta_{\xi}\Psi=\left(\xi^{\mu}\partial_{\mu}-\frac{1}{8}\nabla_{\mu}\xi_{\nu}[\gamma^{\mu},\gamma^{\nu}]\right)\Psi.$
(33)

Evaluating (33) for $\xi=\xi_{Y}$ on (14) one finds

$\delta_{\xi_{Y}}\Psi=\frac{1}{2^{1/4}r}\begin{pmatrix}-\bar{z}\delta_{Y}\psi_{-}\\\
\delta_{Y}\psi_{-}\\\ \delta_{Y}\psi_{+}\\\
z\delta_{Y}\psi_{+}\end{pmatrix}+\cdots$ (34)

where

$\displaystyle\delta_{Y}\psi_{-}$ $\displaystyle=$
$\displaystyle\left(Y(z)\partial_{z}+\frac{1}{2}Y^{\prime}(z)+\frac{1}{2}Y^{\prime}(z)u\partial_{u}\right)\psi_{-}$
(35) $\displaystyle\delta_{Y}\psi_{+}$ $\displaystyle=$
$\displaystyle\left(Y(z)\partial_{z}+Y^{\prime}(z)+\frac{1}{2}Y^{\prime}(z)u\partial_{u}\right)\psi_{+}$
(36)

Similarly, for infinitesimal Lorentz transformations $\xi_{\bar{Y}}$
parametrized by $\bar{Y}(\bar{z})\partial_{\bar{z}}$ one finds

$\displaystyle\delta_{\bar{Y}}\psi_{-}$ $\displaystyle=$
$\displaystyle\left(\bar{Y}(\bar{z})\partial_{\bar{z}}+\bar{Y}^{\prime}(\bar{z})+\frac{1}{2}\bar{Y}^{\prime}(\bar{z})u\partial_{u}\right)\psi_{-}$
(37) $\displaystyle\delta_{\bar{Y}}\psi_{+}$ $\displaystyle=$
$\displaystyle\left(\bar{Y}(\bar{z})\partial_{\bar{z}}+\frac{1}{2}\bar{Y}^{\prime}(\bar{z})+\frac{1}{2}\bar{Y}^{\prime}(\bar{z})u\partial_{u}\right)\psi_{+}.$
(38)

From a 2d perspective, the above expressions determine the
holomorphic/antiholomorphic conformal dimensions of the ($u$-independent part
of) $\psi_{\pm}$. In particular, one concludes the 2d spin (or equivalently
the 4d helicity) of $\psi_{\pm}$ is equal to $\pm 1/2$.

The analogous calculation on the gauge fields leads to

$\displaystyle\delta_{Y}A_{z}$ $\displaystyle=$
$\displaystyle\left(Y(z)\partial_{z}+Y^{\prime}(z)+\frac{1}{2}Y^{\prime}(z)u\partial_{u}\right)A_{z}$
(39) $\displaystyle\delta_{Y}A_{\bar{z}}$ $\displaystyle=$
$\displaystyle\left(Y(z)\partial_{z}+\frac{1}{2}Y^{\prime}(z)u\partial_{u}\right)A_{\bar{z}}$
(40) $\displaystyle\delta_{\bar{Y}}A_{z}$ $\displaystyle=$
$\displaystyle\left(\bar{Y}(\bar{z})\partial_{\bar{z}}+\frac{1}{2}\bar{Y}^{\prime}(\bar{z})u\partial_{u}\right)A_{z}$
(41) $\displaystyle\delta_{\bar{Y}}A_{\bar{z}}$ $\displaystyle=$
$\displaystyle\left(\bar{Y}(\bar{z})\partial_{\bar{z}}+\bar{Y}^{\prime}(\bar{z})+\frac{1}{2}\bar{Y}^{\prime}(\bar{z})u\partial_{u}\right)A_{\bar{z}}$
(42)

from which one concludes the 2d spin/4d helicity of $A_{z}$ is $+1$ and that
of $A_{\bar{z}}$ is $-1$.

For holomorphic vector fields, the generator is given by

$J_{Y}=\sum_{s=\pm}\int_{\mathcal{I}}dud^{2}x\left(\dot{A}_{-s}\delta_{Y}A_{s}+i\bar{\psi}_{s}\delta_{Y}\psi_{s}\right),$
(44)

with similar expression holding in the antiholomorphic case. These represent
the total angular momentum of the system evaluated at null infinity.

####  II.5.2 Translations

Spacetime translations (or more generally supertranslations) take the
asymptotic form near null infinity

$\xi_{f}=f(z,\bar{z})\partial_{u}+\cdots$ (45)

where the sphere function $f$ for a translation $a^{\mu}$ is

$f(z,\bar{z})=a_{\mu}q^{\mu}(z,\bar{z})$ (46)

with $q^{\mu}$ given in (9). The action on the asymptotic fields is obtained
from the Lie derivative, leading to

$\delta_{f}\psi_{s}=f\dot{\psi}_{s},\quad\delta_{f}A_{s}=f\dot{A}_{s}.$ (47)

The phase space generator of the above transformation is then given by

$P_{f}=\sum_{s=\pm}\int_{\mathcal{I}}dud^{2}x\left(\dot{A}_{-s}\delta_{f}A_{s}+i\bar{\psi}_{s}\delta_{f}\psi_{s}\right),$
(48)

and represents the total linear momentum of the system evaluated at null
infinity.

####  II.5.3 Large $U(1)$ gauge

Gauge transformations (7) with asymptotic behavior

$\Lambda(r,u,x)\stackrel{{\scriptstyle r\to\infty}}{{=}}\lambda(x)+\cdots$
(49)

induce the following action on the fields at null infinity:

$\delta_{\lambda}A_{a}=\partial_{a}\lambda,\quad\delta_{\lambda}\psi_{s}=-ie\lambda\psi_{s}.$
(50)

The canonical generator for (50) is given by

$Q_{\lambda}=\int_{\mathcal{I}}dud^{2}x\lambda\big{(}-\partial^{a}\dot{A}_{a}+e(\bar{\psi}_{+}\psi_{+}+\bar{\psi}_{-}\psi_{-})\big{)}.$
(51)

By expressing the fermionic field in terms of Fock operators, the “hard” part
of the charge can be written as

$Q^{\textrm{hard}}_{\lambda}=e\int
d^{2}x\lambda(\rho^{\psi}_{+}+\rho^{\psi}_{-}-\rho^{\bar{\psi}}_{+}-\rho^{\bar{\psi}}_{-}).$
(52)

For $\lambda=1$, $Q_{\lambda}$ reduces to the total electric charge as
measured at null infinity:

$Q_{\lambda=1}=e(N^{\psi}_{+}+N^{\psi}_{-})-e(N^{\bar{\psi}}_{+}+N^{\bar{\psi}}_{-}),$
(53)

where we recall that $N^{X}_{\pm}$ is the number operator defined in (30).

####  II.5.4 Axial rotations

The axial symmetry (8) induces at null infinity the transformation

$\delta_{\text{A}}\psi_{s}=-is\psi_{s}.$ (54)

The canonical generator is given by

$Q_{\text{A}}=(N^{\psi}_{+}+N^{\bar{\psi}}_{+})-(N^{\psi}_{-}+N^{\bar{\psi}}_{-})$
(55)

and counts the excess of positive helicity fermions over negative helicity
fermions.

##  III Soft electrons and fermionic asymptotic charges

In this section we present a tree-level formula for soft electrons (and soft
positrons) and compute the associated asymptotic charges. For simplicity we
treat all particles as outgoing, and correspondingly derive the asymptotic
charge at future null infinity. A detailed derivation of the soft electron
theorem is given in appendix B.

Consider a tree-level amplitude involving $n$ hard particles and a soft
electron of momentum $p^{\mu}=\omega q^{\mu}(x)$, with $\omega\to 0$ and
$q(x)$ as in (9). As in other instances of soft theorems, the dominant
diagrams are those where the soft electron is attached to an external hard
leg. There are two possibilities. Either the soft electron emerges from an
external hard photon, leaving behind an internal electron line, or it emerges
from an external hard positron, leaving behind an internal photon line:

or

Figure 1: The two types of diagrams contributing to a soft electron amplitude.

In both cases one obtains a result that is proportional to the $n$-point
amplitude left behind the soft emission vertex. However, unlike the situation
for soft photons, there is a change in the type of hard particle involved in
the process.888A change in the type of hard particles also occurs for soft
photinos stromfermionic  and for subleading soft photons in presence of non-
minimally coupled matter elvang2 ; mitra . The proportionality factor in the
two types of process coincide, and is trivial unless there is certain helicity
matching at the vertex. Taking for concreteness a soft electron of positive
helicity and calling $p_{i}$ the momentum of the hard particle, the
proportionality factor is given by (see appendix B)

$-e\frac{\bar{u}_{+}(p)\not{\varepsilon}^{+}_{i}u_{+}(p_{i})}{2p\cdot
p_{i}}=\frac{e}{\sqrt{\omega\omega_{i}}(z-z_{i})}.$ (56)

One thus arrives at the (positive helicty) soft electron theorem

$\displaystyle\mathcal{A}_{n+1}(\\{p_{i}\\},\omega
q^{\psi}_{+}))\stackrel{{\scriptstyle\omega\to 0}}{{=}}$
$\displaystyle\frac{e}{\sqrt{\omega}}\sum_{i\in
A_{+}}\frac{1}{\sqrt{\omega_{i}}(z-z_{i})}\mathcal{A}_{n}(\ldots,{p_{i}}^{A}_{+}\to{p_{i}}^{\psi}_{+},\dots)$
(57) $\displaystyle+$
$\displaystyle\frac{e}{\sqrt{\omega}}\sum_{i\in\bar{\psi}_{-}}\frac{1}{\sqrt{\omega_{i}}(z-z_{i})}\mathcal{A}_{n}(\ldots,{p_{i}}^{\bar{\psi}}_{-}\to{p_{i}}^{A}_{-},\dots),$

where we use the labels $\psi,\bar{\psi}$ and $A$ for electrons, positrons and
photons respectively and particle helicities are displayed by $\pm$
subscripts. The argument in $\mathcal{A}_{n}$ indicates the $n$-point
amplitude involves a change in the $i$-th hard particle type.

Following what has been done for other soft theorems, we can interpret the
above formula as a Ward identity of an asymptotic charge. Since the particle
states in (57) are normalized according to the standard momentum-space Fock
operators (see appendix A), it is simpler to first write the charge in terms
of them. From (57) one can read-off the direction-dependent fermionic charge

$\frac{4\pi i}{\sqrt{2}}Q_{\psi_{+}}(x):=\lim_{\omega\to
0}\sqrt{\omega}b^{standard}_{+}(\omega q(x))\\\
-e\int\widetilde{dp^{\prime}}\frac{1}{\sqrt{\omega^{\prime}}(z-z^{\prime})}\left(a^{standard\,\dagger}_{+}(p^{\prime})b^{standard}_{+}(p^{\prime})+\
c^{standard\,\dagger}_{-}(p^{\prime})a^{standard}_{-}(p^{\prime})\right).$
(58)

The overall normalization is chosen for later convenience and the label
_standard_ is to distinguish the momentum-space Fock operators from the ones
defined in section II.4.

Next, we look to express the charge in terms of the asymptotic fields
$\psi_{s}(u,x)$ and $A_{s}(u,x)$. To achieve this, we first rewrite (58) in
terms of the asymptotic Fock operators of section II.4. These are related to
the momentum-space operators by (see appendix A)

$a^{standard}_{s}=4\pi ia_{s},\quad b^{standard}_{s}=\frac{4\pi
i}{\sqrt{2\omega}}b_{s},\quad c^{standard}_{s}=\frac{4\pi
i}{\sqrt{2\omega}}c_{s}.$ (59)

Substituting (59) in (58) and using
$\widetilde{dp}\,^{\prime}=\frac{\omega^{\prime}}{2(2\pi)^{3}}d^{2}x^{\prime}d\omega^{\prime}$
we get

$Q_{\psi_{+}}(x)=\lim_{\omega\to 0}b_{+}(\omega,x)\\\ +\frac{ie}{2\pi}\int
d^{2}x^{\prime}\frac{1}{(z-z^{\prime})}\int_{0}^{\infty}\frac{d\omega^{\prime}}{2\pi}\left(a^{\dagger}_{+}(\omega^{\prime},x^{\prime})b_{+}(\omega^{\prime},x^{\prime})+\
c^{\dagger}_{-}(\omega^{\prime},x^{\prime})a_{-}(\omega^{\prime},x^{\prime})\right).$
(60)

We now Fourier transform from $\omega$ to $u$-space using the expressions from
section II.4. Let us discuss the two terms in (60) separately. Following the
standard terminology, we refer to them as “soft” and “hard” charge
respectively. The “soft” part of the charge is found to be given by

$Q^{soft}_{\psi_{+}}(x)\equiv\lim_{\omega\to
0}b_{+}(\omega,x)=\int_{-\infty}^{\infty}du\psi_{+}(u,x).$ (61)

The hard term is a bit more involved. We start by rewriting it as

$Q^{hard}_{\psi_{+}}(x)=ie\partial^{-1}_{\bar{z}}\sigma_{\bar{z}+}(x),$ (62)

where

$\partial^{-1}_{\bar{z}}=\frac{1}{2\pi}\int
d^{2}x^{\prime}\frac{1}{(z-z^{\prime})},$ (63)

and

$\sigma_{\bar{z}+}(x):=\int_{0}^{\infty}\frac{d\omega^{\prime}}{2\pi}\left(a^{\dagger}_{+}(\omega^{\prime},x^{\prime})b_{+}(\omega^{\prime},x^{\prime})+\
c^{\dagger}_{-}(\omega^{\prime},x^{\prime})a_{-}(\omega^{\prime},x^{\prime})\right).$
(64)

From Eqs. (23) and (25) one finds (64) can be written as

$\displaystyle\sigma_{\bar{z}+}(x)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{d\omega^{\prime}}{2\pi}\left(\tilde{A}_{\bar{z}}(-\omega^{\prime},x^{\prime})\tilde{\psi}_{+}(\omega^{\prime},x^{\prime})+\tilde{\psi}_{+}(-\omega^{\prime},x^{\prime})\tilde{A}_{\bar{z}}(\omega^{\prime},x^{\prime})\right)$
(65) $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}duA_{\bar{z}}(u,x^{\prime})\psi_{+}(u,x^{\prime}).$
(66)

We finally combine the “soft” and “hard” charges by factoring out an inverse
derivative in the former

$Q^{soft}_{\psi_{+}}(x)=\partial^{-1}_{\bar{z}}\int_{-\infty}^{\infty}du\partial_{\bar{z}}\psi_{+}(u,x).$
(67)

Comparing (67) with (62) and (66), we conclude the total charge can be written
as

$Q_{\psi_{+}}(x)=\partial^{-1}_{\bar{z}}\int^{\infty}_{-\infty}duD_{\bar{z}}\psi_{+}(u,x),$
(68)

where

$D_{a}\psi_{s}\equiv(\partial_{a}+ieA_{a})\psi_{s}$ (69)

is the gauge covariant derivative at null infinity.

Repeating the previous analysis for a negative helicity soft electron yields a
charge of the form

$Q_{\psi_{-}}(x)=\partial^{-1}_{z}\int^{\infty}_{-\infty}duD_{z}\psi_{-}(u,x).$
(70)

Finally, soft positrons lead to charges that are the complex conjugates of
(68) and (70).

It is natural to combine all these direction-dependent charges into a single
smeared asymptotic charge which we define as

$F_{\chi}:=i\int
d^{2}xdu\left(\bar{\chi}^{\bar{z}}_{+}D_{\bar{z}}\psi_{+}+\bar{\chi}^{z}_{-}D_{z}\psi_{-}+\chi^{z}_{+}D_{z}\bar{\psi}_{+}+\chi^{\bar{z}}_{-}D_{\bar{z}}\bar{\psi}_{-}\right).$
(71)

where $\chi^{z}_{+}(x)$ and $\chi^{\bar{z}}_{-}(x)$ are the components of a
spinor-vector smearing parameter

$\chi=(\chi^{z}_{+},\chi^{\bar{z}}_{-}),$ (72)

with complex conjugate
$\bar{\chi}=(\bar{\chi}^{\bar{z}}_{+},\bar{\chi}^{z}_{-})$. We take $\chi$ to
be grassmanian so that $F_{\chi}$ is real/hermitian.

##  IV On fermionic asymptotic symmetries

Even though conserved asymptotic charges imply the existence of asymptotic
symmetries, the nature of the latter may be challenging to decipher (see for
example stromLow ; stromfermionic ). One reason for this difficulty is that
asymptotic symmetries are spontaneously broken stromlectures , while the
charges obtained from soft theorems are evaluated on a single vacuum sector.
To characterize the symmetry one needs to make manifest the vacuum manifold,
thus going beyond standard Fock-space amplitudes Kapec:2017tkm ; Choi:2017ylo
; Prabhu:2022zcr ; Kapec:2022hih ; allthat . At the classical level, this
usually requires an extension of the radiative phase space, see e.g.
Kapec:2014zla ; Freidel:2019ohg ; perazanagy ; Peraza:2023ivy ;
Sudhakar:2023uan ; Aggarwal:2018ilg ; Campiglia:2020qvc ; Capone:2023roc ;
Chowdhury:2022gib .

In this section we take the first steps towards characterizing the asymptotic
fermionic symmetry implied by the soft electron theorem. We will start by
evaluating the charge action according to the radiative phase space brackets
(21). We shall see this action does not respect the $|u|\to\infty$ behavior of
the asymptotic fields, thus indicating the need of a phase space extension. We
leave for future work the identification of such extension as well as the
related problem of fully characterizing the symmetry algebra. We will
nevertheless be able to verify the commutator between fermionic and bosonic
symmetries. In appendix C we shall further discuss a naive evaluation of the
commutator between two fermionic symmetries.

###  IV.1 Symmetry action from radiative PBs

Infinite dimensional phase spaces may present subtleties that are absent in
finite dimensions. In particular, not all phase-space functionals are
guaranteed to yield well-defined PB actions. As we shall see, this is the case
for the fermionic charge $F_{\chi}$.

Let us for a moment ignore the aforementioned subtlety and consider the action
obtained from the standard formula

$\delta_{\chi}=\\{\cdot,F_{\chi}\\}.$ (73)

From the elementary PBs (21) and the expression for $F_{\chi}$ (71) one
finds999We recall that $D_{a}=\partial_{a}+ieA_{a}$ is the gauge covariant
derivative at null infinity and
$\chi_{z-}=\chi^{\bar{z}}_{-},\quad\chi_{\bar{z}+}=\chi^{z}_{+}$. The action
on $\bar{\psi}_{\pm}$ can be obtained from that of $\psi_{\pm}$ by complex
conjugation. To simplify expressions we have chosen to display
$\delta_{\chi}\dot{A}_{s}$ rather than $\delta_{\chi}A_{s}$ (which is non-
local in $u$).

$\displaystyle\delta_{\chi}\psi_{+}$ $\displaystyle=D_{z}\chi^{z}_{+}$ (74)
$\displaystyle\delta_{\chi}\psi_{-}$
$\displaystyle=D_{\bar{z}}\chi^{\bar{z}}_{-}$
$\displaystyle\delta_{\chi}\dot{A}_{z}$
$\displaystyle=\frac{e}{2}(\bar{\chi}_{z+}\psi_{+}-\chi_{z-}\bar{\psi}_{-})$
$\displaystyle\delta_{\chi}\dot{A}_{\bar{z}}$
$\displaystyle=\frac{e}{2}(\bar{\chi}_{\bar{z}-}\psi_{-}-\chi_{\bar{z}+}\bar{\psi}_{+}).$

We first notice that $\lim_{u\to\pm}\delta_{\chi}\psi_{s}\neq 0$, and thus the
transformation does not preserve the condition
$\psi_{s}\stackrel{{\scriptstyle|u|\to\infty}}{{\to}}0$ (20). This suggests a
phase space extension that allows for non-trivial asymptotic values of
$\psi_{s}$ when $u\to\pm\infty$.

The transformation rule for $\dot{A}_{s}$ (74) is compatible with the
asymptotic behaviour of $A_{s}$ given in (19) provided $\psi_{s}$ decays to
zero as in (20). The discussion from the previous paragraph however indicates
that in an extended space where $\lim_{u\to\pm\infty}\psi_{s}\neq 0$ we would
need to allow for non-trivial $|u|\to\infty$ values of $\dot{A}_{s}$. This
resembles the situation for the asymptotic charges obtained from subleading
photons stromLow , whose action creates an $O(u)$ term in $A_{s}$.

If one continues the previous considerations back and forth between
$\delta_{\chi}\psi_{s}$ and $\delta_{\chi}A_{s}$, one is led to conclude that
all powers of $u$ should be allowed in the $|u|\to\infty$ behaviour of the
fields. We recall however that (74) is at best valid only around the trivial
vacuum and cannot be trusted beyond linear order in $\chi$. Higher order
iterations of $\delta_{\chi}$ may require the inclusion of terms that are
absent in (74) and that could modify the previous conclusion. We leave for
future work the elucidation of this non-linear structure.101010A preliminary
attempt go beyond linear order is presented in appendix C, where we evaluate
the commutator between two fermionic variations (74). Although the result
cannot be trusted due to the aforementioned limitations, it may still be of
use in more complete treatments. There are however checks to be made at first
order in $\chi$, namely the commutation of $\delta_{\chi}$ with the bosonic
symmetries reviewed in section II.5. We discuss them next.

###  IV.2 Algebra relations with bosonic symmetries

Let $\delta_{B}$ be a bosonic symmetry action corresponding to a charge $B$,
i.e.

$\delta_{B}=\\{\cdot,B\\}.$ (75)

We restrict attention to the bosonic symmetries discussed in section II.5 so
that $\delta_{B}=\delta_{Y},\delta_{f},\delta_{\lambda},\delta_{\text{A}}$ for
$B=J_{Y},P_{f},Q_{\lambda},Q_{\text{A}}$ respectively. As we shall see, these
symmetries have a natural action on the fermionic parameter,

$\chi\mapsto\chi+\delta_{B}\chi$ (76)

such that the commutator of variations is given by

$[\delta_{\chi},\delta_{B}]=\delta_{\delta_{B}\chi}.$ (77)

At the level of charges, this should imply the PB relations111111In the
absence of a central extension, as it turns out to be the case.

$\\{F_{\chi},B\\}=-F_{\delta_{B}\chi}.$ (78)

There are however various subtleties in the evaluation of PBs that make (78)
challenging to verify directly. It is for this reason that we will instead
focus on the relations

$\displaystyle\delta_{B}F_{\chi}$ $\displaystyle=$ $\displaystyle-
F_{\delta_{B}\chi},$ (79) $\displaystyle\delta_{\chi}B$ $\displaystyle=$
$\displaystyle F_{\delta_{B}\chi}.$ (80)

It turns out that (79) follows straightforwardly from the expression of
$F_{\chi}$ and (77). This should lead, via (75), to (78). Finally from (73)
one would arrive at (80). There are however two obstructions to this logic
chain.

First, Eq. (75) does not hold for $Q_{\lambda}$ if one uses the radiative PBs
(21) (see the discussion following this equation). Since these were the
brackets used in the definition of $\delta_{\chi}$ (74), there will be a
mismatch in (80) when $B=Q_{\lambda}$. This problem should go away if
$\delta_{\chi}$ is constructed from improved PBs as the ones proposed in
stromQED .

A second obstruction appears from the fact that the relation

$\delta F_{\chi}=\Omega(\delta,\delta_{\chi})$ (81)

only holds for variations $\delta$ that decay to zero faster than the assumed
ones in order to compensate for the singular behavior of $\delta_{\chi}$ at
$u=\pm\infty$. This leads to additional difficulties in verifying (80), for
instance when $B=J_{Y}$. A complete fix to this second obstruction would
presumably require the inclusion of soft fermionic degrees of freedom in the
symplectic structure.

We now discuss in more detail the situation for each bosonic symmetry
separately.

####  IV.2.1 Lorentz

The transformation properties of $\chi$ under the Lorentz group follow
directly from those of the elementary fields $A_{s}$ and $\psi_{\pm}$. By
direct evaluation one readily obtains (77) and (79) with

$\displaystyle\delta_{(Y,\bar{Y})}\chi^{z}_{+}$
$\displaystyle=\left(Y\partial_{z}+\bar{Y}\partial_{\bar{z}}+\frac{1}{2}\bar{Y}^{\prime}\right)\chi^{z}_{+}$
(82) $\displaystyle\delta_{(Y,\bar{Y})}\chi^{\bar{z}}_{-}$
$\displaystyle=\left(Y\partial_{z}+\bar{Y}\partial_{\bar{z}}+\frac{1}{2}Y^{\prime}\right)\chi^{\bar{z}}_{-}$
. (83)

Thus, from a 2d perspective, $\chi^{z}_{+}$ and $\chi^{\bar{z}}_{-}$ have
(anti-)holomorphic dimensions $(h,\bar{h})=(0,1/2)$ and $(1/2,0)$
respectively.

Relation (80) is verified provided one discards boundary terms of the type

$\left[A_{\bar{z}}\delta_{\chi}\delta_{(Y,\bar{Y})}A_{z}\right]^{u=\infty}_{u=-\infty},$
(84)

which are however non-trivial under the assumed fall-offs (19), (20).

####  IV.2.2 Translations

It is easy to verify that $\delta_{\chi}$ commutes with translations,

$[\delta_{\chi},\delta_{f}]=0$ (85)

and that $\delta_{f}F_{\chi}=0$ (assuming the fall-offs (19), (20) otherwise
there could be non-zero boundary terms). The relation $\delta_{\chi}P_{f}=0$
follows with no caveats.

####  IV.2.3 Large $U(1)$ gauge

As in the Lorentz case, the transformation properties of $\chi$ follow from
those of the elementary fields. One can easily verify (77) and (79) with

$\displaystyle\delta_{\lambda}\chi^{z}_{+}$ $\displaystyle=$ $\displaystyle-
ie\chi^{z}_{+}$ (86) $\displaystyle\delta_{\lambda}\chi^{\bar{z}}_{-}$
$\displaystyle=$ $\displaystyle-ie\chi^{\bar{z}}_{-}.$ (87)

As anticipated earlier, Eq. (80) is not verified unless one addresses the
$1/2$ discontinuity in the radiative PBs (22). Let us describe how the issue
arises in the present context.

The RHS of (80) takes the form

$\displaystyle F_{\delta_{\lambda}\chi}$ $\displaystyle=$
$\displaystyle-e\int_{\mathcal{I}}\lambda\bar{\chi}^{\bar{z}}_{+}D_{\bar{z}}\psi_{+}+\cdots$
(88) $\displaystyle=$
$\displaystyle-e\int_{\mathcal{I}}\partial_{\bar{z}}\lambda\bar{\chi}^{\bar{z}}_{+}\psi_{+}+e\int_{\mathcal{I}}\lambda
D_{\bar{z}}\bar{\chi}^{\bar{z}}_{+}\psi_{+}+\cdots,$ (89)

where in the second line we integrated by parts and for simplicity we are only
displaying what corresponds to the first term in $F_{\chi}$ (71) (the others
follow a similar pattern).

The LHS of (80) can be written as a

$\delta_{\chi}Q_{\lambda}=\delta_{\chi}Q^{\textrm{hard}}_{\lambda}+\delta_{\chi}Q^{\textrm{soft}}_{\lambda}$
(90)

with

$\displaystyle\delta_{\chi}Q^{\textrm{hard}}_{\lambda}$ $\displaystyle=$
$\displaystyle\delta_{\chi}\int_{\mathcal{I}}e\lambda(\bar{\psi}_{+}\psi_{+}+\bar{\psi}_{-}\psi_{-})$
(91) $\displaystyle=$ $\displaystyle
e\int_{\mathcal{I}}\lambda\delta_{\chi}\bar{\psi}_{+}\psi_{+}+\cdots$ (92)
$\displaystyle=$ $\displaystyle e\int_{\mathcal{I}}\lambda
D_{\bar{z}}\bar{\chi}^{\bar{z}}_{+}\psi_{+}+\cdots$ (93)

and

$\displaystyle\delta_{\chi}Q^{\textrm{soft}}_{\lambda}$ $\displaystyle=$
$\displaystyle\int_{\mathcal{I}}(\partial_{\bar{z}}\lambda\dot{A}_{z}+\partial_{z}\lambda\dot{A}_{\bar{z}})$
(94) $\displaystyle=$
$\displaystyle\int_{\mathcal{I}}\partial_{\bar{z}}\lambda\delta_{\chi}\dot{A}_{z}+\cdots$
(95) $\displaystyle=$
$\displaystyle\frac{e}{2}\int_{\mathcal{I}}\partial_{\bar{z}}\lambda\bar{\chi}^{\bar{z}}_{+}\psi_{+}+\cdots,$
(96)

where we only displayed the terms that correspond to those shown in (89).
Comparing the expressions one sees that whereas
$\delta_{\chi}Q^{\textrm{hard}}_{\lambda}$ reproduces the first term in (89),
$\delta_{\chi}Q^{\textrm{soft}}_{\lambda}$ fails to reproduce the second term
by a factor of $1/2$. The origin of this mismatch is the same as the one
leading to the discontinuity discussed in Eq. (22) and so it would be fixed by
the use of improved PBs stromQED .

####  IV.2.4 Axial rotations

In this case, equations (79) and (80) are verified with no caveats, with

$\displaystyle\delta_{\text{A}}\chi^{z}_{+}$ $\displaystyle=$
$\displaystyle-i\chi^{z}_{+}$ (97)
$\displaystyle\delta_{\text{A}}\chi^{\bar{z}}_{-}$ $\displaystyle=$
$\displaystyle i\chi^{\bar{z}}_{-}.$ (98)

##  V Outlook

Over the past decade there has been a fruitful revision on the subject of
asymptotic symmetries in (asymptotically) flat spacetimes, driven by the
discovery of their connection with soft theorems and memory effects
stromlectures . Whereas asymptotic symmetries were originally tied to gauge
symmetries that are non-trivial at infinity, their relation to soft theorems
led to a broader perspective. Indeed, there is a growing list of soft theorems
that admit an interpretation in terms of asymptotic symmetries with no obvious
gauge origin.121212What appears as a non-gauge asymptotic symmetry may
sometimes be realized as large gauge, either by allowing divergent gauge
transformations subphotonAL  or by performing a change of field variables
ronak . It is however not clear whether such reinterpretation is always
possible. Here we have enlarged such list, by presenting an asymptotic
symmetry associated to soft electrons in tree-level massless
QED.131313Although we have set our discussion in the context of QED, it should
admit a straightforward generalization to the case of non-abelian gauge fields
as well as chiral fermions. Following what is by now a standard procedure, we
identified the form of the asymptotic charge and initiated the study of its
symmetry action.

To our knowledge, this is the first example of a fermionic asymptotic symmetry
in a theory with no _standard_ supersymmetry. The existence of such fermionic
symmetry, however, suggest that tree-level massless QED possess an
_asymptotic_ supersymmetry algebra. Unfortunately the present work remains
agnostic as to what such algebra should be. Below we describe this and others
shortcomings of our analysis as well as possible strategies to overcome them.

We followed what may be referred to as a canonical approach to asymptotic
symmetries, in which the classical phase space at null infinity provides the
bridge between symmetries and charges. A difficulty with this approach is that
it often requires an enlarged version of the radiative phase space, with no
simple recipe to determine it.141414Recent developments on so called corner
symmetries donnellyfreidel ; geiller ; Freidel:2021dxw ; leigh  may in fact
yield such recipe (at least in the case of large gauge symmetries). Whereas we
showed that an enlargement is implied by soft electrons, we were not able to
characterize it beyond linear level. This in turn impeded us to trustfully
evaluate the commutator between fermionic asymptotic symmetries.

There is however a second approach in which symmetries are described in terms
of 2d conserved currents of a dual CFT Barnich:2013axa ; stromYM ; sundrum .
There, the algebraic structure is read off from collinear factorization
theorems TaylorOPE ; Taylor2 ; stromOPE ; holosymalg ; OPEallspins ; jiang ;
shamik1 ; shamik2  without the need of phase-space considerations.151515The
price to pay, however, is that so far such approach only captures symmetry
relations among generators of the same helicity sign. See freidelhs ;
freidelYM ; Donnay:2018neh ; puhm ; Crawley:2021ivb  for discussions on how
the canonical and CFT approaches relate to each other. It would be very
interesting to study the fermionic symmetries presented here from this
perspective, as it could allow us to extract information on the fermionic
symmetry algebra.

Regardless of the approach, it has been found in the context of gauge and
gravitational theories that completeness of the asymptotic symmetry algebra
requires charges associated to subleading soft theorems of arbitrarily high
order, see e.g. clspatial ; freidelhs . It would be interesting to explore the
existence of higher order soft electron theorems and their connection with the
soft photon ones. A hint that a nontrivial interplay may occur is provided by
a naive evaluation of the commutator between the fermionic variations
presented here, leading to an expression that is reminiscent to the subleading
soft photon charge action (see appendix C).

A final pressing open problem regards the fate of the symmetries beyond tree-
level. The structure of their modification or breaking due to loop corrections
should be worth studying. In particular, it would be interesting to explore
any possible connection with the chiral anomaly, whose consequences at null
infinity were recently analyzed in agullo .

## Acknowledgements

We would like to thank Iván Agulló, Alok Laddha, Guzmán Hernández, Pablo Pais
and Michael Reisenberger for illuminating discussions. We specially thank Alok
Laddha for his feedback and encouragement. AA acknowledges support from
PEDECIBA and from CSIC grant I+D 583. MC acknowledges support from PEDECIBA
and from ANII grant FCE-1-2019-1-155865. This research was supported in part
by Perimeter Institute for Theoretical Physics. Research at Perimeter
Institute is supported by the Government of Canada through the Department of
Innovation, Science and Economic Development Canada and by the Province of
Ontario through the Ministry of Research, Innovation and Science.

##  Appendix A Asymptotic vs. momentum-space Fock operators

In this appendix we work out the relation between the Fock operators
introduced in section II.4 and the standard momentum-space Fock operators. In
particular we verify they yield equivalent (anti) commutation relations.

We start with the momentum expansion of asymptotic free fields:

$\displaystyle\mathcal{A}_{\mu}(X)$
$\displaystyle=\sum_{s=\pm}\int\widetilde{dp}\,\left(a_{s}(p)\varepsilon^{s*}_{\mu}(p)e^{ip\cdot
X}+a^{\dagger}_{s}(p)\varepsilon^{s}_{\mu}(p)e^{-ip\cdot X}\right)$ (99)
$\displaystyle\Psi(X)$
$\displaystyle=\sum_{s=\pm}\int\widetilde{dp}\,\left(b_{s}(p)u_{s}(p)e^{ip\cdot
X}+c^{\dagger}_{s}(p)v_{s}(p)e^{-ip\cdot X}\right)$

where $\widetilde{dp}\,=d^{3}p/((2\pi)^{3}2|p|)$ and the photon/electron
polarization vectors are described below. To simplify notation we do not yet
include additional labels to the Fock operators, but it should be kept in mind
that they have different normalization from those introduced in section II.4.

To study the null infinity limit of these expressions, we parametrize
$X^{\mu}$ in terms of $(r,u,x)$ as in (11) and write the null momentum
$p^{\mu}$ as

$p^{\mu}=\omega q^{\mu}(x^{\prime})$ (100)

with $q^{\mu}$ as in (9) with $x^{\prime}=(z^{\prime},\bar{z}^{\prime})$. In
this parametrization the momentum measure takes the form

$\widetilde{dp}\,=\frac{\omega}{2(2\pi)^{3}}d^{2}x^{\prime}d\omega$ (101)

and the plane-wave phase becomes

$p\cdot X=r\omega q(x)\cdot q(x^{\prime})+u\omega k\cdot q(x^{\prime})$ (102)

where

$q(x)\cdot q(x^{\prime})=-|z-z^{\prime}|^{2}.$ (103)

In the $r\to\infty$ limit, the $d^{2}x^{\prime}$ integral can be evaluated by
a saddle point analysis. In the simplest case of a positive-frequency scalar
one gets

$\lim_{r\to\infty}\int\widetilde{dp}\,\phi(p)e^{ip\cdot X}=\frac{1}{4\pi
ir}\int^{\infty}_{0}\frac{d\omega}{2\pi}\phi(\omega q(x))e^{-i\omega
u}+O(1/r^{2}).$ (104)

Before extending this formula to the photon and fermion fields, let us recall
the form of the corresponding momentum wave-functions. The polarization vector
for a photon with momentum $p=\omega q^{\mu}(x)$ can be taken to be stromQED

$\varepsilon^{+\mu}=\partial_{z}q^{\mu}=\frac{1}{\sqrt{2}}(\bar{z},1,-i,-\bar{z}),\quad\varepsilon^{-\mu}=\partial_{\bar{z}}q^{\mu}=\frac{1}{\sqrt{2}}(z,1,i,-z)$
(105)

which satisfy

$\varepsilon^{\pm}\cdot
q=0,\quad\varepsilon^{\pm}\cdot\varepsilon^{\pm}=0,\quad\varepsilon^{+}\cdot\varepsilon^{-}=1,\quad\varepsilon^{+}_{\mu}\varepsilon^{-}_{\nu}+\varepsilon^{-}_{\mu}\varepsilon^{+}_{\nu}+\frac{q_{\mu}k_{\nu}+k_{\mu}q_{\nu}}{q\cdot
k}=\eta_{\mu\nu}$ (106)

with $k^{\mu}$ a null vector with non-zero dot product with $q^{\mu}$, as for
instance the one given in (10). For the fermion we have (see e.g. elvang ;
stromfermionic )

$u_{+}=v_{-}=2^{1/4}\sqrt{\omega}\begin{pmatrix}0\\\ 0\\\ 1\\\
z\end{pmatrix},\quad
u_{-}=v_{+}=2^{1/4}\sqrt{\omega}\begin{pmatrix}-\bar{z}\\\ 1\\\ 0\\\
0\end{pmatrix}$ (107)

which satisfy

$\not{q}u_{\pm}=0,\quad\bar{u}_{s}\gamma^{\mu}u_{s^{\prime}}=2p^{\mu}\delta_{ss^{\prime}},\quad\sum_{s}u_{s}\bar{u}_{s}=-\not{p},$
(108)

and similarly for $v_{\pm}$. Expressions (105), (107) can be obtained from
those used in the spinor-helicity formalism elvang  with the choice of spinor
$2^{1/4}\sqrt{\omega}(1,z)$ for the momentum $p^{\mu}=\omega
q^{\mu}(z,\bar{z})$.161616The photon polarization in (105) corresponds to a
choice of “reference spinor” of the form $(1,\infty)$.

Using the above expressions, one finds

$\mathcal{A}_{z}=\frac{1}{4\pi
i}\int^{\infty}_{0}\frac{d\omega}{2\pi}\left(a_{+}e^{-i\omega
u}-a^{\dagger}_{-}e^{i\omega u}+\right)+O(1/r)$ (109) $\Psi=\frac{1}{4\pi
ir}\int^{\infty}_{0}\frac{d\omega}{2\pi}2^{1/4}\sqrt{\omega}\begin{pmatrix}\begin{array}[]{c}-\bar{z}\\\
1\end{array}(b_{-}e^{-i\omega u}-c^{\dagger}_{+}e^{i\omega u})\\\
\begin{array}[]{c}1\\\ z\end{array}(b_{+}e^{-i\omega
u}-c^{\dagger}_{-}e^{i\omega u})\end{pmatrix}.$ (110)

Comparing with (23), (25) we see that the momentum space Fock operators are
related to the Fock operators of section II.4 by

$a_{s}=\frac{1}{4\pi i}a^{standard}_{s},\quad b_{s}=\frac{\sqrt{2\omega}}{4\pi
i}b^{standard}_{s},\quad c_{s}=\frac{\sqrt{2\omega}}{4\pi i}c^{standard}_{s},$
(111)

where we have added the label “standard” to the operators of this section to
distinguish them from those of section II.4. The commutation relations (26)
then imply the standard momentum-space (anti) commutators

$[d^{standard}_{s}(p),d^{standard\,\dagger}_{s^{\prime}}(p^{\prime})]=2|p|(2\pi)^{3}\delta_{ss^{\prime}}\delta^{(3)}(p-p^{\prime}),\quad
d=\\{a,b,c\\},$ (112)

where we used that
$\delta^{(3)}(p-p^{\prime})=\frac{1}{\omega^{2}}\delta(\omega-\omega^{\prime})\delta^{(2)}(x,x^{\prime})$
for $p=\omega q(x)$ and $p^{\prime}=\omega q(x^{\prime})$.

##  Appendix B Soft photons and soft electrons

In this appendix we derive the tree-level formulas for the emission of soft
particles. We treat all particles as outgoing.

Since the formulas are dictated by the QED 3-point interaction, it will be
useful to start the discussion by recalling the structure of general 3-point
amplitudes. Consider then a process involving an electron, positron and photon
of momenta

$\displaystyle p_{e}$ $\displaystyle=\omega_{e}q(z_{e},\bar{z}_{e})$ (113)
$\displaystyle p_{\bar{e}}$
$\displaystyle=\omega_{\bar{e}}q(z_{\bar{e}},\bar{z}_{\bar{e}})$
$\displaystyle p_{\gamma}$
$\displaystyle=\omega_{\gamma}q(z_{\gamma},\bar{z}_{\gamma})$

and helicities $s,\bar{s}$ and $h$. The (momentum conserving delta function
stripped) amplitude is given by

$A_{3}((p_{e},s),(p_{\bar{e}},\bar{s}),(p_{\gamma},h))=\includegraphics[valign={m}]{A_3}=-ie\bar{u}_{s}(p_{e})\gamma^{\mu}v_{\bar{s}}(p_{\bar{e}})\varepsilon^{h}_{\mu}(p_{\gamma}).$
(114)

In order for the amplitude to be non-zero we must have $\bar{s}=-s$ whereas
$h$ can be arbitrary. Writing the 4 posibilites in holomorphic coordinates
(using the expressions for polarization vectors given in the previous section)
one finds

$\displaystyle A_{3}((p_{e},+),(p_{\bar{e}},-),(p_{\gamma},+))$
$\displaystyle=2ie\sqrt{\omega_{e}\omega_{\bar{e}}}(\bar{z}_{\gamma}-\bar{z}_{e})$
(115) $\displaystyle A_{3}((p_{e},+),(p_{\bar{e}},-),(p_{\gamma},-))$
$\displaystyle=2ie\sqrt{\omega_{e}\omega_{\bar{e}}}(z_{\gamma}-z_{\bar{e}})$
$\displaystyle A_{3}((p_{e},-),(p_{\bar{e}},+),(p_{\gamma},+))$
$\displaystyle=2ie\sqrt{\omega_{e}\omega_{\bar{e}}}(\bar{z}_{\gamma}-\bar{z}_{\bar{e}})$
$\displaystyle A_{3}((p_{e},-),(p_{\bar{e}},+),(p_{\gamma},-))$
$\displaystyle=2ie\sqrt{\omega_{e}\omega_{\bar{e}}}(z_{\gamma}-z_{e}).$

We have not yet imposed momentum conservation. This typically leads to trivial
amplitudes (unless one allows for complex momenta elvang ). An exception
however is when one of the particles goes soft with the remaining two becoming
coincident:171717Since we are working with real momenta, $z\to z_{0}$ implies
$\bar{z}\to\bar{z}_{0}$.

$\displaystyle\omega_{\gamma}\to 0$ $\displaystyle\implies z_{\bar{e}}\to
z_{e},\quad\omega_{\bar{e}}\to\omega_{e}$ (116) $\displaystyle\omega_{e}\to 0$
$\displaystyle\implies z_{\bar{e}}\to
z_{\gamma},\quad\omega_{\bar{e}}\to\omega_{\gamma}$
$\displaystyle\omega_{\bar{e}}\to 0$ $\displaystyle\implies z_{e}\to
z_{\gamma},\quad\omega_{e}\to\omega_{\gamma}.$

Notice that in the soft electron/positron limit, two out of the four
amplitudes in (115) vanish.

We next describe the behavior of general amplitudes when one particle goes
soft. As is well known, the leading contribution comes from diagrams where the
soft particle is attached to an external hard particle. In the following we
discuss the relevant diagrams for each type of soft particle. We consider
processes involving $n$ hard particles and $1$ soft particle.

###  B.1 Soft photon

The amplitude for a process in which a photon is attached to an external
electron is given by

$\displaystyle=-ie\varepsilon_{\mu}^{h}(p_{\gamma})\bar{u}_{s}(p_{e})\gamma^{\mu}i\frac{\not{p}_{e}+\not{p}_{\gamma}}{(p_{e}+p_{\gamma})^{2}}\text{\bf
A}_{n-1}(p_{e}+p_{\gamma})$ (117)
$\displaystyle\stackrel{{\scriptstyle\omega_{\gamma}\to
0}}{{=}}e\varepsilon_{\mu}^{h}(p_{\gamma})\bar{u}_{s}(p_{e})\gamma^{\mu}\frac{\not{p}_{e}}{2p_{e}\cdot
p_{\gamma}}\text{\bf A}_{n-1}(p_{e})+O(\omega_{\gamma}^{0})$
$\displaystyle=-e\frac{\varepsilon^{h}(p_{\gamma})\cdot p_{e}}{p_{\gamma}\cdot
p_{e}}A_{n}+O(\omega_{\gamma}^{0})$

where $\text{\bf A}_{n-1}$ denotes the spinor-valued off-shell amplitude
corresponding to the remaining $n-1$ hard particles. In going to the second
line we kept the leading terms in the $\omega_{\gamma}\to 0$ limit, whereas in
going to the third line we used the identities in (108) and the fact that the
fermion helicities in the vertex must match for the amplitude to be nonzero.
$A_{n}$ represents the amplitude for the $n$ hard particles.

One can similarly compute the amplitude for the case where the photon is
attached to an outgoing positron, resulting in an expression as the above
modulo an overall sign. Summing over all external particles one obtains
Weinberg’s soft photon theorem weinberg .

###  B.2 Soft electron

We now consider processes in which an electron goes soft. From the structure
of the QED vertex, we see that the electron can be attached either to an
external photon or to an external positron (recall we are taking all particles
to be outgoing).

The contribution when the electron is attached to an external photon is

$\displaystyle=-ie\varepsilon_{\mu}^{h}(p_{\gamma})\bar{u}_{s}(p_{e})\gamma^{\mu}i\frac{\not{p}_{e}+\not{p}_{\gamma}}{(p_{e}+p_{\gamma})^{2}}\text{\bf
A}_{n-1}(p_{e}+p_{\gamma})$ (118)
$\displaystyle\stackrel{{\scriptstyle\omega_{e}\to
0}}{{=}}e\varepsilon_{\mu}^{h}(p_{\gamma})\bar{u}_{s}(p_{e})\gamma^{\mu}\frac{\not{p}_{\gamma}}{2p_{e}\cdot
p_{\gamma}}\text{\bf A}_{n-1}(p_{\gamma})+O(\omega_{e}^{0})$
$\displaystyle=-e\frac{\varepsilon_{\mu}^{h}(p_{\gamma})\bar{u}_{s}(p_{e})\gamma^{\mu}u_{s}(p_{\gamma})}{2p_{e}\cdot
p_{\gamma}}A_{n}((\gamma,p_{\gamma},h)\to(e,p_{\gamma},s))+O(\omega_{e}^{0}).$

Notice that the starting point is the same as in the soft photon case
discussed in (117). The difference is that we are now taking the electron,
rather than the photon, to be soft. Again one obtains an expression that is
proportional to an $n$-point amplitude $A_{n}$. However, unlike the soft
photon theorem, the external hard particle involved in the process is replaced
by a different type of particle (in this case a photon is replaced by an
electron, as indicated in the argument of $A_{n}$). The multiplicative factor
in front of $A_{n}$ can be simplified by writing the momenta and polarization
vectors in holomorphic coordinates. The only non-trivial case is when $h$ and
$s$ are of the same sign (this can be seen from Eqs. (115) and (116)) and one
finds

$\displaystyle-e\frac{\varepsilon_{\mu}^{+}(p_{\gamma})\bar{u}_{+}(p_{e})\gamma^{\mu}u_{+}(p_{\gamma})}{2p_{e}\cdot
p_{\gamma}}$ $\displaystyle=$
$\displaystyle\frac{e}{\sqrt{\omega_{e}\omega_{\gamma}}(z_{e}-z_{\gamma})},$
(119)
$\displaystyle-e\frac{\varepsilon_{\mu}^{-}(p_{\gamma})\bar{u}_{-}(p_{e})\gamma^{\mu}u_{-}(p_{\gamma})}{2p_{e}\cdot
p_{\gamma}}$ $\displaystyle=$
$\displaystyle\frac{e}{\sqrt{\omega_{e}\omega_{\gamma}}(\bar{z}_{e}-\bar{z}_{\gamma})}.$
(120)

The contribution when the electron is attached to an external positron is

$\displaystyle=-ie\bar{u}_{s}(p_{e})\gamma^{\mu}v_{-s}(p_{\bar{e}})\frac{-i\eta_{\mu\nu}}{(p_{e}+p_{\bar{e}})^{2}}\text{A}^{\nu}_{n-1}(p_{e}+p_{\bar{e}})$
(121) $\displaystyle\stackrel{{\scriptstyle\omega_{e}\to
0}}{{=}}-e\bar{u}_{s}(p_{e})\gamma^{\mu}u_{s}(p_{\bar{e}})\frac{\sum_{h}\varepsilon^{h}_{\mu}(p_{\bar{e}})\varepsilon^{-h}_{\nu}(p_{\bar{e}})}{2p_{e}\cdot
p_{\bar{e}}}\text{A}^{\nu}_{n-1}(p_{\bar{e}})+O(\omega_{e}^{0})$
$\displaystyle=-e\frac{\varepsilon_{\mu}^{s}(p_{\bar{e}})\bar{u}_{s}(p_{e})\gamma^{\mu}u_{s}(p_{\bar{e}})}{2p_{e}\cdot
p_{\bar{e}}}A_{n}((\bar{e},p_{\bar{e}},-s)\to(\gamma,p_{\bar{e}},-s))+O(\omega_{e}^{0}).$

Here we assumed the positron helicity is opposite to that of the soft
electron, since otherwise the amplitude vanishes. $\text{A}^{\nu}_{n-1}$ is a
vector-valued off-shell amplitude associated to the $n-1$ hard particles not
involved in the process under consideration. In going to the second line we
used the resolution of the identity (106) together with $\not{p}u(p)=0$ and
$p_{\nu}\text{A}^{\nu}_{n-1}(p)=0$. Finally, in the third line, we used the
fact that only the $h=s$ photon polarization yields a non-zero contribution.

We can finally combine both type of contributions to obtain the “soft electron
theorem” of Eq. (57).

##  Appendix C Naive commutator of fermionic charges and subleading soft
photons

In this appendix we present the asymptotic symmetry associated to Low’s
subleading photon theorem Low , as first introduced by Lysov, Pasterski and
Strominger (LPS) stromLow . We next evaluate the commutator of two fermionic
symmetries and compare it with the LPS symmetry. Even though the two
expression do not coincide, there are certain similarities that suggest the
LPS symmetry may feature in an eventual (super) symmetry algebra involving the
fermionic generators. We hope the calculation presented here will be of use in
a more complete treatment.

###  C.1 Asymptotic symmetries for subleading soft photon

Soft photons obey a factorization theorem at subleading order Low  that can be
understood in terms of LPS asymptotic symmetries stromLow . The corresponding
charges are parametrized by vector fields $Y^{a}$ on the sphere and take the
form,

$Q^{\text{LPS}}_{Y}=\int_{\mathcal{I}}dud^{2}x\left(-2u\dot{A}_{\bar{z}}\partial_{z}^{2}Y^{z}+u\partial_{z}Y^{z}j_{u}+Y^{z}j_{z}\right)+c.c.,$
(122)

where $j_{u}$ and $j_{a}$ are the leading components of the asymptotic
current,

$\displaystyle\mathcal{J}_{u}$ $\displaystyle=$ $\displaystyle
j_{u}/r^{2}+\cdots$ (123) $\displaystyle\mathcal{J}_{a}$ $\displaystyle=$
$\displaystyle j_{a}/r^{2}+\cdots.$ (124)

In order to compute the symmetry action on the spinor field, we need to
express $j_{u}$ and $j_{a}$ in terms of $\psi_{s}$. This is achieved by
substituting the asymptotic expansion of the Dirac spinor (14) in the current
expression (3). For the $u$ component one gets

$j_{u}=-e\sum_{s}\bar{\psi}_{s}\psi_{s}.$ (125)

The evaluation of $j_{a}$ requires one order further in the asymptotic
expansion of the Dirac field,

$\Psi=\frac{1}{r}\overset{0}{\Psi}+\frac{1}{r^{2}}\overset{1}{\Psi}+\cdots$
(126)

where $\overset{0}{\Psi}$ is the leading term displayed in (14). Substituting
(126) in the current expression one finds

$\displaystyle\mathcal{J}_{a}$ $\displaystyle=$
$\displaystyle\frac{e}{r}\left(\overset{0}{\overline{\Psi}}+\frac{1}{r}\overset{1}{\overline{\Psi}}+\cdots\right)\partial_{a}\not{q}\left(\overset{0}{\Psi}+\frac{1}{r}\overset{1}{\Psi}+\cdots\right)$
(127) $\displaystyle=$
$\displaystyle\frac{e}{r^{2}}\left(\overset{0}{\overline{\Psi}}\partial_{a}\not{q}\overset{1}{\Psi}+\overset{1}{\overline{\Psi}}\partial_{a}\not{q}\overset{0}{\Psi}\right)+\cdots$
(128)

where we used that

$\overset{0}{\overline{\Psi}}\partial_{a}\not{q}\overset{0}{\Psi}=0.$ (129)

We finally need to express $\overset{1}{\Psi}$ in terms of $\psi_{s}$.
Imposing the asymptotic free-field equation on $\Psi$ one finds181818Whereas
this suffices for tree-level considerations, a more complete treatment should
deal with the full non-linear asymptotic field equations. This in turn may
modify the assumed fall-offs (126) (and hence (123), (124)). See sayali  for
related discussions.

$0=\not{\partial}\Psi=-\frac{1}{r}\not{q}\partial_{u}\overset{0}{\Psi}+\\\
\frac{1}{r^{2}}\left(-\not{q}\partial_{u}\overset{1}{\Psi}+\not{k}\overset{0}{\Psi}+\partial_{a}\not{q}\partial^{a}\overset{0}{\Psi}\right)+\cdots$
(130)

The vanishing of the $1/r$ term leads to (14), which we write as191919In (14)
and (131) we are setting to zero a possible $u$-independent spinor that may
not be in the kernel of $\not{q}$. It may be that such spinor is needed in an
extended version of the radiative phase space.

$\overset{0}{\Psi}=\psi_{+}\mathbf{u}_{+}+\psi_{-}\mathbf{u}_{-}$ (131)

where

$\mathbf{u}_{+}=\frac{1}{2^{1/4}}\begin{pmatrix}0\\\ 0\\\ 1\\\
z\end{pmatrix},\quad\mathbf{u}_{-}=\frac{1}{2^{1/4}}\begin{pmatrix}-\bar{z}\\\
1\\\ 0\\\ 0\end{pmatrix},$ (132)

span the kernel of the matrix $\not{q}$.

$\overset{1}{\Psi}$ is to be determined from the vanishing of the $1/r^{2}$
term in (130). Due to the non-invertibility of $\not{q}$, it is not
immediately obvious this equation can be solved. However, using that
$k^{\mu}=1/2\partial^{a}\partial_{a}q^{\mu}$, the equation can be brought into
the form

$0=-\not{q}(\partial_{u}\overset{1}{\Psi}+\frac{1}{2}\partial^{a}\partial_{a}\overset{0}{\Psi})+\frac{1}{2}\partial^{a}\partial_{a}(\not{q}\overset{0}{\Psi}).$
(133)

Since the last term vanishes due to (131), the equation fixes
$\overset{1}{\Psi}$ modulo elements in the kernel of $\not{q}$:202020The first
term in (134) is what one gets from solving $\square\Psi=0$. The apparent
indeterminacy in $\overset{1}{\Psi}$ gets fixed upon requiring integrability
on the equation for $\overset{2}{\Psi}$. However, we do not need such terms
for the present discussion.

$\displaystyle\overset{1}{\Psi}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\partial_{u}^{-1}\partial^{a}\partial_{a}\overset{0}{\Psi}+\text{ker}(\not{q}).$
(134) $\displaystyle=$
$\displaystyle-\sum_{s}\partial_{u}^{-1}\partial_{a}\psi_{s}\partial^{a}\mathbf{u}_{s}+\text{ker}(\not{q}).$
(135)

Substituting in (127) we get

$\displaystyle j_{z}$ $\displaystyle=$
$\displaystyle-e\left(\bar{\psi}_{-}\partial_{z}\partial_{u}^{-1}\psi_{-}+\partial_{z}\partial_{u}^{-1}\bar{\psi}_{+}\psi_{+}\right),$
(136) $\displaystyle j_{\bar{z}}$ $\displaystyle=$
$\displaystyle-e\left(\bar{\psi}_{+}\partial_{\bar{z}}\partial_{u}^{-1}\psi_{+}+\partial_{\bar{z}}\partial_{u}^{-1}\bar{\psi}_{-}\psi_{-}\right),$
(137)

where we used that
$\bar{\mathbf{u}}_{s}\partial_{a}\not{q}\mathbf{u}_{s^{\prime}}=0$ (which
ensures $j_{a}$ is independent of the $\text{ker}(\not{q})$ indeterminacy in
$\overset{1}{\Psi}$), together with the fact that the only nonzero components
of
$\bar{\mathbf{u}}_{s}\partial_{a}\not{q}\partial_{b}\mathbf{u}_{s^{\prime}}$
are

$\bar{\mathbf{u}}_{+}\partial_{\bar{z}}\not{q}\partial_{z}\mathbf{u}_{+}=\bar{\mathbf{u}}_{-}\partial_{z}\not{q}\partial_{\bar{z}}\mathbf{u}_{-}=1.$
(138)

Defining $\delta^{\text{LPS}}_{Y}=\\{\cdot,Q^{\text{LPS}}_{Y}\\}$ with the PBs
(21) one obtains

$\displaystyle\delta^{\text{LPS}}_{Y}\psi_{+}$
$\displaystyle=ie\left(Y^{a}\partial_{a}+u\partial_{a}Y^{a}\partial_{u}+\partial_{z}Y^{z}\right)\partial_{u}^{-1}\psi_{+},$
(139) $\displaystyle\delta^{\text{LPS}}_{Y}\psi_{-}$
$\displaystyle=ie\left(Y^{a}\partial_{a}+u\partial_{a}Y^{a}\partial_{u}+\partial_{\bar{z}}Y^{\bar{z}}\right)\partial_{u}^{-1}\psi_{-},$
$\displaystyle\delta^{\text{LPS}}_{Y}\dot{A}_{z}$
$\displaystyle=-\partial_{z}^{2}Y^{z}.$

###  C.2 Naive commutator of two fermionic generators

In this section we evaluate the (anti) commutator between two fermionic
variations $\delta_{\chi}$ defined in (74). As discussed there, one should not
take this result too seriously, since the expressions do not incorporate
“corner” fermionic degrees of freedom that are presumably needed to have a
well defined phase space action.

To facilitate the computation, let us introduce the notation

$\chi_{+}=\chi_{z-},\quad\chi_{-}=\chi_{\bar{z}+}$ (140)

so that (74) can be written more compactly as,

$\displaystyle\delta_{\chi}\psi_{s}$ $\displaystyle=D_{s}\chi_{-s},$ (141)
$\displaystyle\delta_{\chi}\dot{A}_{s}$
$\displaystyle=\frac{e}{2}\left(\bar{\chi}_{-s}\psi_{s}-\chi_{s}\bar{\psi}_{-s}\right).$

We focus on the anticommutator
$[\delta_{\chi},\delta_{\chi}]=2\delta_{\chi}\delta_{\chi}$, since the general
case can be obtained from this basic one. Acting twice with (141) one finds,

$\displaystyle\psi_{+}$
$\displaystyle=-ie^{2}\left(\bar{\chi}_{-}\chi_{-}\partial_{u}^{-1}\psi_{+}-\chi_{+}\chi_{-}\partial^{-1}_{u}\bar{\psi}_{-}\right)$
(142) $\displaystyle[\delta_{\chi},\delta_{\chi}]\dot{A}_{z}$
$\displaystyle=-e\left(D_{z}\bar{\chi}_{+}\chi_{+}+\bar{\chi}_{-}D_{z}\chi_{-}\right)$

(the corresponding expressions for $\psi_{-}$ and $\dot{A}_{\bar{z}}$ can be
obtained from (142) by interchanging $+\leftrightarrow-$ and
$z\leftrightarrow\bar{z}$). The result bears certain resemblance with
$\delta^{\text{LPS}}_{Y}$ (139) if

$Y^{z}\sim e\partial_{z}^{-1}\bar{\chi}_{s}\chi_{s}.$ (143)

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