New Symmetries of Massless QED

An infinite number of physically nontrivial symmetries are found for abelian gauge theories with massless charged particles. They are generated by large $U(1)$ gauge transformations that asymptotically approach an arbitrary function $\varepsilon(z,\bar{z})$ on the conformal sphere at future null infinity ($\mathscr I^+$) but are independent of the retarded time. The value of $\varepsilon$ at past null infinity ($\mathscr I^-$) is determined from that on $\mathscr I^+$ by the condition that it take the same value at either end of any light ray crossing Minkowski space. The $\varepsilon\neq$ constant symmetries are spontaneously broken in the usual vacuum. The associated Goldstone modes are zero-momentum photons and comprise a $U(1)$ boson living on the conformal sphere. The Ward identity associated with this asymptotic symmetry is shown to be the abelian soft photon theorem.

in the conventional vacuum. The soft photons appear as Goldstone modes living on the sphere at the boundary of I .
These large U(1) gauge symmetries are precise analogs of BMS supertranslations in gravity. 1 It is curious that discovery of the gravitational symmetry preceded its electromagnetic analog by a half-century.
The relation between soft theorems and asymptotic symmetries of I + (but not of the S-matrix), was described already in [1], which in turn was inspired by [27]. Two "simplifying" restrictions were made in the analysis of [1]: the incoming state was required to be invariant under the large gauge symmetries, and the parameter ε(z,z) was required to be locally holomorphic. However, far from simplifying the analysis, these restrictions obscured the underlying structure. The present analysis both simplifies and generalizes that of [1]. This paper considers theories in which there are no stable massive charged particles, and the quantum state begins and ends in the vacuum at past and future timelike infinity. Of course, in realworld QED the electron is a stable massive charged particle, so it is highly desirable to generalize our analysis to this case. 2 However, stable massive charges create technical complications because the charge current has no flux through future null infinity. Rather, there is charge flux across timelike infinity which becomes a singular point in the conformal compactification of Minkowski space. In principle a systematic treatment of this singularity should be possible -the fields disperse and are weakly interacting; nonetheless, this is well beyond the scope of the present paper. This paper is organized as follows. In section 2 we describe the classical final data formulation at I + . Section 3 gives the asymptotic symmetries which are the QED analogs of BMS symmetries and constructs the associated charges. In section 4 the commutators at I + are given and the charges of section 3 are shown to generate the symmetries. This requires a careful treatment of the Goldstone modes and boundary conditions at the boundaries of I + . Section 5 gives the corresponding formulae for I − . In section 6 we give conditions which tie the data of I − to that of I + and thereby defines the scattering problem. The conditions are shown to break the separate asymptotic symmetries to a diagonal subgroup preserving the S-matrix. In section 7 the quantum Ward identity of this symmetry is shown to relate scattering amplitudes with and without a soft photon insertion. Finally in section 8 we show that this Ward identity is the soft photon theorem.
2 Asymptotic expansion at I + In this subsection we consider the canonical final data formulation of U(1) electrodynamics coupled to massless charged matter at future null infinity (I + ). It is convenient to adopt retarded coordinates 1 It would be interesting to systematically derive this large U (1) symmetry group using the type of asymptotic analysis employed in BMS [22]: herein the non-triviality of the symmetries is established by their equivalence to the soft photon theorem. 2 The present analysis is relevant to hard scattering in QED when the electron mass becomes negligible.
where u = t − r and γ zz = 2 (1+zz) 2 is the round metric on the conformal sphere. 3 Thus I + is the null S 2 × R boundary at r = ∞ with coordinates (u, z,z). I + has boundaries at u = ±∞, which we denote I + ± . The bulk equations of motion for U(1) gauge theory are We work in the retarded radial gauge We wish to expand the fields around I + . The radiation flux through I + is proportional to To ensure that the radiation flux is nonzero and finite, we require A z ∼ O(1) near I + .
Following (2.6), we also require A u ∼ O(1/r) near I + , giving the expansion (2.7) The leading terms in the field strengths near I + are then Note that the fields F and A live on I + and have no r dependence. The leading constraint equation where j u (u, z,z) = lim r→∞ r 2 j M u (u, r, z,z) . (2.10) We will be interested in configurations which revert to the vacuum in the far future, i.e.
From (2.9) and (2.11) we can determine A u in terms of A z and Az (for a given j u ), which we will take to be coordinates on the asymptotic phase space Γ + . Subleading terms in the 1 r expansions of all the other equations of motion then determine the expansion of A in terms of the final data A z and matter current.
The analogous structure at I − is described in section 5 below.

Large gauge transformations
The gauge conditions (2.5) and (2.6) leave unfixed residual gauge transformations generated by an arbitrary function approachingε = ε(z,z) on the conformal sphere at r = ∞. We will refer to these as "large gauge transformations". The action on Γ + is These comprise the asymptotic symmetries considered in this paper. The associated charge is [23] Q + ε = 1 e 2 In the second equality we have integrated by parts, assumed the final charge relaxes to zero at I + + and used the constraint (2.9). For the special case ε = 1, Q + 1 is the total initial electric charge which obeys For the choice ε(z,z) = δ 2 (z − w) one has the fixed-angle charge This is the total outgoing electric charge radiated into the fixed angle (w,w) on the asymptotic S 2 .
The first term is a linear "soft" photon (by which we mean momentum is strictly zero, as opposed to just small) contribution to the fixed-angle charge. It does not contribute to the total charge Q + 1 as it is a total derivative. The second term is the accumulated matter charge flux at the angle (w,w).
Q + ε generates the large gauge transformation on matter fields where Φ is any massless charged matter field operator on I + with charge q.

Canonical formulation
The commutators, or equivalently a non-degenerate symplectic form, on the radiative phase space [23,28]. The non-vanishing ones are Integrating and fixing the integration constants by antisymmetry gives where Θ(x) = sign(x). Given (3.5), one might expect that symmetry transformations on the gauge fields are then generated by commutators with Q + ε using (4.2). However, an explicit computation gives which is off by a factor of 1 2 . A similar factor of 1 2 was encountered in the construction of the BMS supertranslation operator in [3].
In order to resolve this discrepancy, we must give a more precise description of the phase space Γ + . In particular we must both specify boundary conditions on A z at the boundaries I + ± of I + and include the soft photon zero modes. The boundary values of the fields are denoted by We consider here the sector of the phase space with no long-range magnetic fields, namely In other words, the connections A ± z are flat on I + ± . We will implement (4.5) as constraints. These constraints are not preserved by the commutators (4.2), which hence must be modified according to Dirac's procedure. A unique modification is determined by the continuity condition and the vanishing of equal-u commutators. The solution to (4.5) is Of course, the constant modes of φ ± cannot be determined from A ± z , but it natural and useful to include them by simply treating φ ± (z,z) as unconstrained fields on S 2 . The commutators satisfying (4.6) are then (4.8) Using (4.7), the charge Q + ε can be written as It then immediately follows from (4.8) and (4.9) that (4.10) Moreover, the charges satisfy the Abelian algebra Hence, on the constrained phase space defined by (4.5) the modified commutators properly generate the large gauge transformations.
Periodicity of ε implies that φ − lives on a circle of radius 1 e 2 : and have (in our conventions) integer charges n. Such operators do not in themselves create physical states. Rather states with charge n are created by products of these operators with neutral mattersector operators. This is virtually the same operator product decomposition familiar in 2D CFT when factoring a U(1) current algebra boson, or in 4D soft collinear effective field theory (SCET) involving the so-called jet field [29,30].
A vacuum wave function for the Goldstone mode which we take to be φ − can be defined by the condition φ − (z,z)|0 = 0. (4.14) (4.10) implies that the large gauge symmetries are broken in this vacuum. The symmetries transform (4.14) into more general φ − eigenstates obeying Up to an undetermined normalization, the inner products are z)) . i.e. zero frequency mode duF uz = e 2 ∂ z (φ + − φ − ), is orthogonal to this form and has no symplectic partner among these radiative modes. We remedy this by adding the boundary degree of freedom A − z and constructing a symplectic form which pairs it with this zero mode. This is done consistently with the constraint (4.5) on both A − z and the conjugate zero mode representing the absence of long range magnetic fields. The resulting non-degenerate phase space Γ + = {F uz (u, z,z), φ + (z,z), φ − (z,z)} then consists of the usual (non-zero frequency) radiative modes, together with the zero-momentum soft photon φ + (z,z) − φ − (z,z) and the canonically conjugate periodic real boson φ − (z,z).
5 Asymptotic structure at I − A similar structure exists near I − and is needed to discuss scattering. In this subsection we recap the requisite formulae. I − is at r = ∞ with v fixed in advanced coordinates ds 2 = −dv 2 + 2dvdr + 2r 2 γ zz dzdz, The leading order constraint equation is where now Unfixed large gauge transformations are parameterized by ε − (z,z) under which The associated charge is As on I + , we define the boundary values of the fields and impose constraints This is solved by Employing the same methods and assumptions as our analysis near I + , the commutators consistent [ψ + (z,z), ψ − (w,w)] = i 4πe 2 log |z − w| 2 .

(5.10)
These in turn imply The incoming phase space is then Γ The classical scattering problem is to find the map from Γ − to Γ + , i.e. to determine the final data (F uz , φ − ) on I + which arises from a given set of initial data (G vz , ψ + ) on I − . Given a field strength everywhere on Minkowski space, this data is so far determined only up to the large gauge transformations which are generated by both ε and ε − and act separately on Γ + and Γ − .
Clearly there can be no sensible scattering problem without imposing a relation between ε and ε − . Any relation between them should preserve Lorentz invariance. Under an SL(2, C) Lorentz transformation parameterized by ζ z ∼ 1, z, z 2 one finds This symmetry is preserved by the natural requirement ψ + (z,z) = φ − (z,z) . However in the quantum theory, where phases matter, they have significant consequences to which we now turn.

Quantum Ward identity
In this section, we consider the consequences of the large gauge symmetry on the semi-classical S-matrix. Let us denote an in (out) state comprised of n (m) particles with charges q in k (q out k ), incoming at points z in k (outgoing at points z out k ) on the conformal sphere S 2 by |in ≡ z in 1 , · · · , z in n ( out| ≡ z out 1 , · · · , z out m |). The S-matrix elements are then denoted as out| S |in . The quantum version of the classical invariance of scattering under large gauge transformations is The semi-classical charge obeys the quantum relations (from (3.2) and (5.6)) This Ward identity relates the insertion of a soft photon with polarization and normalization given in (7.4) into any S-matrix element to the same S-matrix element without a soft photon insertion.
For an incoming state which happens to be the vacuum, (7.2) reduces to Hence Q − ε does not annihilate the vacuum unless ε = constant, implying that all but the constant mode of the large gauge symmetries are spontaneously broken. Moreover (5.11) identifies ψ + as the corresponding Goldstone boson.
This result may seem surprising for the following reason. Soft photons are labelled by a spatial direction and a polarization. This suggests two modes for every point on the sphere, which is twice the number predicted by Goldstone's theorem. In fact the positive and negative helicity modes are not independent. As spelled out in Appendix A, there are non-local (on the asymptotic S 2 ) linear combinations of positive and negative helicity photons whose associated soft factor cancels exactly at leading order. 6 To leading order, these linear combinations of soft modes decouple from all S-matrix elements and hence are truly pure gauge. This relation reduces the two modes for every point on the sphere to the single one predicted by Goldstone theorem. Not accounting for this factor of two produced the wrong result in the charge commutator (4.3) and was corrected for in the boundary condition in (4.5).

Soft photon theorem
In this subsection, we show that the Ward identity (7.6) is the soft photon theorem in disguise. In order to do so we must rewrite everything in momentum space. The first step is to expand the soft photon operators F ± [ε], expressed above as weighted integrals over the conformal sphere at I , in terms of the standard plane wave in and out creation and annihilation operators. Momentum eigenmodes in Minkowski space are usually described in flat coordinates related to the retarded coordinates in (2.1) by At late times and large r, the wave packet for a massless particle with spatial momentum centered around p becomes localized on the conformal sphere near the point (z,z) with wherex = x r . The momentum of massless particles may be equivalently characterized either by (ω, z,z) or by p µ . At late times t → ∞, the gauge field A µ becomes free and can be approximated by the mode expansion where q 0 = ω q = | q| and α = ± are the two helicities. The creation and annihilation operators on for ω q > 0 (for ω q = 0 the positive and negative helicities are linearly dependent; see Appendix A). Similarly, a in ± and a in † ± annihilate and create incoming photons on I − . In terms of (w,w) the polarization tensors have components which satisfy q µ ε ±µ ( q) = 0.
To expand the gauge field near I + recall that Using A z = ∂ z x µ A µ , the mode expansion in (8.4) and the stationary phase approximation we find Defining the energy eigenmodes we find When ω > 0 (ω < 0) only the first (second) term contributes. We define the zero mode by the hermitian expression It follows that Similarly on I − we define where a in ± and a in † ± annihilate and create incoming photons on I − . It follows from (8.9) and (8.13) that where F [ε] is defined in (7.4). Setting ε(z,z) = 1 z−w , the Ward identity (7.6) becomes We now wish to compare (8.16) with the soft photon theorem in its conventional form [21] lim

A Decoupled soft photons
It is possible to see directly from the soft photon theorem that a particular combination of positive and negative helicity photons decouples from the theory. This is seen easiest in the (z,z) coordinates.
We start with the soft photon theorem in this parameterization This is precisely the soft photon theorem for a negative-helicity soft photon with momentum pointing towards (z,z). We therefore conclude that the linear combination a out − (ωx) − 1 2π (1 + zz) d 2 w 1 z −w ∂w a out + (ωŷ) 1 + ww (A. 4) has no poles and decouples from the S-matrix at leading order. In the more familiar momentum space variables, this is a out − (ωp γ ) + 1 2π 1 + cos θ pγ dΩ q 1 + cos θ q (ε + (p γ ) ·q) 2 a out + (ωq), (A.5) where the integral is over the angular distribution ofq.