Asymptotic Symmetry, Soft Limits, and the Boundary Interface

Asymptotic symmetries, soft theorems, memory effects, and boundary Ward identities share one charge-balance pattern: an admissible transformation has a charge on each cut, and the change of that charge is balanced by the complete flux through the boundary between the cuts. A quantum Ward identity can encode the same balance. A soft theorem and a memory observable follow only after additional, model-dependent identifications of a zero-frequency radiative mode, asymptotic states, matching data, and endpoint observables. The common charge statement is robust; a universal four-way equivalence is not.

This page makes that interface explicit in four-dimensional Maxwell theory and uses compact Yang–Mills theory only as a bounded contrast. It does not derive a model-specific soft factor, a gravitational memory formula, or a celestial algebra. The research literature and the disagreements described below were checked through 2026-08-03; the conclusions are limited to the cited frameworks and their stated asymptotic assumptions.

Required background. Surface Charges, Integrability, and Ambiguities fixes the charge representative and separates integrability from flux. Boundaries, Flux, and Boundary Ward Identities derives the outward-flux sign and the localized Ward identity.

Helpful background. Charge Algebras, Central Terms, and Corners explains why the parameter bracket, corner representative, and flux prescription must be carried with a charge algebra.

From a finite surface charge to an asymptotic charge

Infinity is not made into an ordinary wall merely by writing a surface integral there. Start instead with a regulator: a large worldtube $B_R$ of radius $R$, cuts $C_R(u)$, a phase space satisfying declared boundary conditions, and a transformation $R_{\epsilon_R}$ that preserves them. The finite-radius surface one-form is

$$\eta_{\epsilon_R,C_R} =\iota_{R_{\epsilon_R}}\Omega_{\Sigma_R}.$$

An asymptotic charge $Q_\epsilon[C]$ exists only if the following operations are controlled as $R\to\infty$:

1. the fields and allowed variations obey falloffs for which $\eta_{\epsilon_R,C_R}$ is finite;
2. $\epsilon_R$ preserves those falloffs and tends to a declared boundary label $\epsilon$;
3. the limiting one-form is locally closed, has vanishing periods on the chosen sector, and therefore integrates to a single-valued charge;
4. the corner improvement and reference normalization have a limit; and
5. the complete lateral presymplectic flux, including any boundary completion, has a finite limit.

The word asymptotic symmetry therefore means a falloff-preserving transformation that survives the null quotient of the limiting presymplectic form. A parameter that is merely nonzero at infinity, or merely residual after gauge fixing, has not yet passed that test. Radiative gravitational boundaries make the distinction especially sharp: a naive Hamiltonian need not exist when symplectic current escapes, and the Wald–Zoupas prescription adds further hypotheses rather than supplying a universal repair (Wald and Zoupas 2000, §§ 2–4, pp. 3–19, Open PDF).

The regulator also blocks a tempting overreach. A finite-distance edge mode does not have a unique, frame-independent soft limit. Here an extrinsic frame is a boundary reference frame constructed from field data in the complementary region, as developed in Edge Modes, Subregions, and Factorization. In Maxwell theory, specific extrinsic frames can relate finite corner charges to asymptotic ones, but different frames probe different cross-boundary data (Araujo-Regado et al. 2025, § 1, pp. 2–7, and § 8, pp. 58–60, v3, Open PDF). The relation is additional structure, not part of the definition of an asymptotic charge.

The common charge-balance statement

Let $C_1$ and $C_2$ be oriented cuts bounding a boundary segment $B_{12}$. For a fixed, field-independent label $\epsilon$, the always-safe comparison of the cut one-forms is

$$\eta_{\epsilon,C_2}-\eta_{\epsilon,C_1} =-\iota_{R_\epsilon} \mathcal W^{\mathrm{comp}}_{B_{12}}.$$

If the contracted complete lateral presymplectic flux is an exact field-space one-form, define

$$\boldsymbol\delta\mathcal F_\epsilon[B_{12}] =\iota_{R_\epsilon} \mathcal W^{\mathrm{comp}}_{B_{12}}.$$

A finite, globally integrable charge with compatible reference choices and a fixed representative then obeys

$$\boxed{ Q_\epsilon[C_2]-Q_\epsilon[C_1] +\mathcal F_\epsilon[B_{12}]=0. }$$

Here $\mathcal F_\epsilon$ is the complete flux in the declared boundary problem. It is not automatically just matter current, radiative energy, or the pullback of a bulk presymplectic current. In the chapter orientation, positive outward $\mathcal F_\epsilon$ lowers the later charge. For a field-dependent label, both the charge variation and flux require the adjusted variation and a rule for transporting the label between cuts.

To turn this classical balance into a scattering statement, one needs future and past asymptotic phase spaces, matched labels, and a scattering operator $\mathsf S$. If every relevant channel has been included and the quantum symmetry is nonanomalous, the matched Ward identity is

$$\begin{aligned} 0={}& \langle \mathrm{out}|\bigl( Q_\epsilon^+\mathsf S -\mathsf S Q_\epsilon^- \bigr)|\mathrm{in}\rangle . \end{aligned}$$

The superscripts $+$ and $-$ refer to future and past asymptotic boundaries; they are not charge signs. The equation is false as written if flux through timelike infinity, a horizon, an internal boundary, or an unmatched corner has simply been dropped.

Suppose, in addition, that a declared infrared prescription permits a split

$$Q_\epsilon^\pm =Q_\epsilon^{\pm,\mathrm H} +Q_\epsilon^{\pm,\mathrm S}.$$

The hard part acts on finite-energy asymptotic excitations. The soft part is linear in a normalized zero-frequency massless mode. Substitution, and nothing more, gives

$$\begin{aligned} &\langle \mathrm{out}|\bigl( Q_\epsilon^{+,\mathrm S}\mathsf S -\mathsf S Q_\epsilon^{-,\mathrm S} \bigr)|\mathrm{in}\rangle \\ &\qquad=-\langle \mathrm{out}|\bigl( Q_\epsilon^{+,\mathrm H}\mathsf S -\mathsf S Q_\epsilon^{-,\mathrm H} \bigr)|\mathrm{in}\rangle . \end{aligned}$$

Only after the soft operator is identified with the correctly normalized $\omega\to0$ creation or annihilation mode, and the hard action with the charges of the external states, does this become a leading soft theorem. For massless four-dimensional QED with standard null-infinity falloffs and antipodal matching, that equivalence is derived explicitly by He et al. 2014, §§ 2–8, corrected v2, printed pp. 2–13, Open PDF.

What each link adds

The same symbols often conceal different mathematical statements. This table keeps the extra input and the stopping point visible.

One charge pattern, five distinct claims

Layer	Statement	Additional input	What does not yet follow
Boundary Ward identity	A localized change of variables relates current divergence, insertions, explicit breaking, anomaly, and wall response.	A regulated measure, transformation-preserving integration domain, boundary conditions, and all contact terms.	No Hamiltonian surface charge, asymptotic state, or soft pole follows from the local identity alone.
Asymptotic charge balance	A finite integrable cut charge changes by minus the complete outward flux.	Falloffs, allowed variations, a preserving parameter, representative, reference, integrability, and every boundary channel.	It supplies neither an in/out scattering operator nor a hard–soft split.
Scattering Ward identity	Matched past and future charges intertwine the scattering operator.	Compatible asymptotic phase spaces, matching through spatial infinity, an infrared state prescription, and no anomaly.	The charge need not contain a normalized one-particle zero mode.
Leading soft theorem	A low-energy massless emission factorizes from a hard process at a declared order.	LSZ or its replacement, mode normalization, a massless pole, hard-charge action, regulator, and an order of limits.	Factorization does not by itself define a classical detector memory or a finite-boundary edge mode.
Memory observable	A detector retains a permanent response determined by an integrated radiative field.	Endpoint limits, a classical or expectation-value regime, detector dynamics, and control of Coulombic and tail terms.	An integrated field is not automatically an exact quantum Ward identity or an all-orders soft theorem.

The logical direction can sometimes be reversed, but only inside a specified model. For example, recovering a Ward identity from a soft theorem requires a complete family of smearing functions and control of the zero mode; recovering a soft theorem from a Ward identity requires the operator and hard-action identifications. Calling the three corners an “infrared triangle” is useful shorthand, not a theorem without hypotheses.

Controlled Maxwell bridge at future null infinity

Consider topologically trivial Maxwell theory in four-dimensional Minkowski spacetime, with no magnetic charges and

$$\partial_\mu F^{\mu\nu}=j^\nu.$$

Use retarded coordinates $u=t-r$ and the site metric convention

$$\mathrm ds^2 =\mathrm du^2+2\,\mathrm du\,\mathrm dr -r^2\gamma_{AB}\mathrm dx^A\mathrm dx^B,$$

where $\gamma_{AB}$ is the unit-sphere metric and $D_A$ its covariant derivative. Future null infinity, denoted $\mathscr I^+$, is approached by $r\to\infty$ at fixed $u$ and angle $\Omega$. Work in a smooth sector with finite integrated flux and falloffs

$$\begin{aligned} F_{ur}&=r^{-2}F_{ur}^{(2)}+O(r^{-3}), &F_{uA}&=F_{uA}^{(0)}+O(r^{-1}),\\ F_{rA}&=O(r^{-2}), &j^r&=r^{-2}j^{r(2)}+O(r^{-3}). \end{aligned}$$

Let the residual parameter tend to a smooth, retarded-time-independent function $\epsilon(\Omega)$. The leading $\nu=r$ Maxwell equation is

$$-\partial_uF_{ur}^{(2)} +D^AF_{uA}^{(0)} =j^{r(2)}.$$

This sign follows directly from the (+---) inverse metric; it is not imported from a conventionally different null-coordinate formula. Define the weighted Coulombic charge and the complete flux on the portion of $\mathscr I^+$ from $u_1$ to $u_2$ by

$$Q_\epsilon(u) =\int_{S^2}\!\mathrm d\Omega\, \epsilon F_{ur}^{(2)},$$ $$\begin{aligned} \mathcal F_\epsilon[u_1,u_2] =\int_{u_1}^{u_2}\!\mathrm du \int_{S^2}\!\mathrm d\Omega\, \bigl( D^A\epsilon\,F_{uA}^{(0)} +\epsilon j^{r(2)} \bigr). \end{aligned}$$

Multiplying the constraint by $\epsilon$, integrating by parts on the sphere, and then integrating in $u$ gives

$$Q_\epsilon(u_2)-Q_\epsilon(u_1) +\mathcal F_\epsilon[u_1,u_2]=0.$$

This is the chapter balance law in an asymptotic regulator limit. For constant $\epsilon$, the radiative term vanishes, and positive outgoing matter current $j^{r(2)}$ lowers the later electric flux. For angle-dependent $\epsilon$, the radiative term is essential; discarding it would manufacture a false conservation law.

The electromagnetic memory one-form is the integrated radiative field

$$\mathcal M_A(\Omega) :=\int_{-\infty}^{+\infty}\!\mathrm du\, F_{uA}^{(0)}(u,\Omega).$$

When the endpoint limits exist with sufficient regularity, retarded radial gauge with

$$A_r=0, \qquad A_u=O(r^{-1}), \qquad A_A=A_A^{(0)}+O(r^{-1})$$

gives $F_{uA}^{(0)}=\partial_uA_A^{(0)}$ and hence

$$\mathcal M_A =A_A^{(0)}(+\infty,\Omega) -A_A^{(0)}(-\infty,\Omega).$$

The radiative contribution to the integrated charge flux is consequently

$$\mathcal F_\epsilon^{\mathrm{rad}} =\int_{S^2}\!\mathrm d\Omega\, D^A\epsilon\,\mathcal M_A.$$

Thus memory and the soft charge can probe the same zero-frequency radiative data. A detector prediction still needs its force law, initial state, endpoint conditions, and any Coulombic subtraction. Pasterski derives the electromagnetic memory relation and its large-$U(1)$ interpretation under explicit null-infinity assumptions (Pasterski 2017, §§ 2–4, pp. 3–11, Open PDF).

Orbit, charge, and radial-gauge descriptions

Three descriptions of the same Maxwell boundary transformation

Description	Classification of ε(Ω)	Invariant content
Gauge-orbit description	Transformations whose boundary pairing vanishes for every allowed variation are quotiented as null. A boundary value is retained only when it preserves the phase space and has a finite nonzero pairing.	“Large” means non-null relative to this boundary problem; it need not mean homotopically nontrivial.
Charge description	Qε is the weighted Coulombic electric flux. Its change is balanced by the radiative and matter terms fixed by the constraint.	The total charge and complete flux are primary; the names hard and soft require the scattering and zero-mode identifications.
Radial-gauge description	Radial gauge leaves a residual parameter ε(Ω). The residual solution is a candidate symmetry, not proof that it is charged.	Setting every residual boundary value to zero changes the boundary quotient; it is not an innocuous completion of gauge fixing.

The three descriptions agree only because the same falloffs, parameter class, allowed variations, and representative were used in each. Changing any one of them requires repeating the classification.

Compact Yang–Mills is not an Abelian substitution

On a finite spatial region, compact Yang–Mills theory has the analogous candidate boundary charge. Choose an invariant Lie-algebra inner product and an orthonormal basis for which the component pairing is $\delta_{ab}$. Then

$$Q_S[\epsilon] =\int_S\!\mathrm dS\, \epsilon^a E^{na},$$

subject to the non-Abelian Gauss constraint $D_iE^{ia}=\rho^a$; this equation defines the sign and normalization of the matter charge density $\rho^a$. Based transformations that vanish at $S$ can be null, while allowed nonzero boundary values can carry charge. In a gauge-fixed description, residual solutions are again only candidates until their boundary pairing, finiteness, and integrability have been checked.

The analogy stops before a componentwise Maxwell interpretation. Under a boundary gauge transformation, both $E^n$ and the Lie-algebra label $\epsilon$ transform in the adjoint representation. Their invariant-inner-product pairing is unchanged when both are transported together, but a fixed color component requires a boundary frame. The constraint and flux are nonlinear, and a hard–soft split need not be separately gauge invariant. At null infinity, perturbative four-dimensional Yang–Mills does admit a leading soft-gluon and current-algebra relation under specific falloffs (Strominger 2014, § 2.4, pp. 11–14, and § 3, pp. 16–17, Open PDF). That result is not an all-orders theorem for compact Yang–Mills on a finite region.

Classical color memory sharpens the qualification. A color flux can produce a nonlinear relative color rotation; only its weak-flux linearization is the Fourier transform of the soft-gluon theorem (Pate, Raclariu, and Strominger 2017, pp. 2–6, Open PDF). Whether a compact Yang–Mills theory is in a confining regime and what nonperturbative asymptotic state space exists are separate questions outside this page. A perturbative colored soft theorem cannot by itself answer them. None of these cautions invalidates the finite-boundary charge; they prevent a verbatim export of the Maxwell infrared triangle.

Where the equivalences stop

Every arrow from charge balance to Ward identity, soft insertion, or memory must survive the following independent checks.

Dimension and asymptotic geometry. The Maxwell calculation above is four-dimensional and asymptotically flat. Radiative and Coulombic orders, tail behavior, and memory change with dimension, especially in odd dimensions (Satishchandran and Wald 2019, §§ II–III, pp. 8–72, corrected v2, Open PDF). AdS, de Sitter, finite timelike walls, and cosmological boundaries require new asymptotic data.

Matching through spatial infinity. Past null infinity $\mathscr I^-$ and future null infinity $\mathscr I^+$ do not share a charge merely because the same symbol $\epsilon$ is written on both. A regularity condition at spatial infinity, denoted $i^0$, must select a matching rule for the parameter and Coulombic data. For Maxwell fields, Prabhu derives charge conservation from such regularity rather than assuming antipodal matching (Prabhu 2018, §§ 3–5, pp. 13–26, v3, Open PDF).

Massive charged sectors. Massive particles reach timelike infinity, not the interior of null infinity. Their hard charge must be supplied on a timelike-infinity phase space. In scalar QED this extension restores the leading Ward/soft correspondence (Campiglia and Laddha 2015, §§ 2–5, pp. 3–18, Open PDF); a null-only formula does not cover massive external charges.

Infrared state space. Charged Fock-state S-matrix elements and the naive Fock scattering construction in four-dimensional QED are infrared divergent or ill-defined. Inclusive observables, coherent dressings, and algebraic formulations answer different questions, and the order of the regulator, energy-resolution, and $\omega\to0$ limits matters. Gabai and Sever show that infrared dressing changes the action and interpretation of large-gauge charges (Gabai and Sever 2016, §§ 3–4, pp. 10–17, substantially revised v2, Open PDF). Hirai and Sugishita construct a gauge-invariant dressed QED scattering operator in which the asymptotic symmetry is implemented differently (Hirai and Sugishita 2021, §§ 2.4–2.6, pp. 9–16, and § 3, pp. 17–18, corrected v2, Open PDF). Prabhu, Satishchandran, and Wald give a stronger counterweight: they identify restricted QED circumstances and an obstruction to the analogous gravitational construction (Prabhu, Satishchandran, and Wald 2022, § 4.4, pp. 49–52, §§ 5.4–5.5, pp. 58–65, and § 6.3, pp. 72–75, v3, Open PDF). Their 2024 follow-up constructs infrared-finite amplitudes between memory sectors, while leaving massless QED and Yang–Mills collinear divergences as a required further generalization (Prabhu and Satishchandran 2024, § 1, pp. 2–8, and § 4.3, pp. 41–42, corrected v3, Open PDF). The page therefore treats the total matched charge as primary and every hard–soft split as prescription-declared.

Endpoints and observables. The integral $\int\mathrm du\,F_{uA}^{(0)}$ exists only with sufficient decay or an explicit distributional prescription. Tails, nonstationary endpoints, magnetic sectors, and detector recoil can change the observable. A formal zero mode is not by itself a measured displacement, kick, phase, or color rotation.

Perturbative order. The clean common statement uses the leading soft mode. Subleading charges, logarithmic terms, loops, anomalies, and renormalized composite operators can alter both sides. A tree-level leading relation cannot be promoted by analogy to an exact all-orders identity.

Theory and boundary channels. Horizons, internal boundaries, defects, and corners may carry charge or flux. Gravity also requires its own radiative phase space and charge prescription. Celestial transforms can reorganize scattering data, but do not establish the underlying Ward/soft/memory relation.

Evidence status and live qualifications

As of 2026-08-03, the cited results support the following conditional synthesis. In the regulated perturbative four-dimensional QED framework of the cited primary derivations, at leading soft order, one must separately specify the massive or massless charge sector, soft and collinear prescriptions, asymptotic states, standard null falloffs, matched past and future data, and controlled endpoints. Within that declared domain, a matched charge Ward identity, the leading soft photon mode, and electromagnetic memory can be linked. The sources above show that massive sectors, dimension, matching, and dressed states are substantive additions rather than notational details. For a current specialist synthesis of asymptotic boundaries, matching, surface charges, and hard–soft fluxes in the gravitational setting, see Donnay 2024, §§ 2.4–3.4, Open PDF v1.

Two broader statements are not supported. First, there is no prescription-independent equivalence for bare Fock, dressed, inclusive, and gravitational scattering (Gabai and Sever 2016, §§ 3–4, pp. 10–17, v2, Open PDF; Hirai and Sugishita 2021, §§ 2.4–3, pp. 9–18, v2, Open PDF; Prabhu, Satishchandran, and Wald 2022, §§ 4–6, pp. 38–75, v3, Open PDF). Second, there is no unique edge-mode-to-soft-mode limit: recent finite-region work makes the relation depend on the chosen extrinsic frame and boundary conditions (Araujo-Regado et al. 2025, § 1, pp. 2–7, and § 8, pp. 58–60, v3, Open PDF). These limitations are reasons to state the charge balance more carefully, not evidence against asymptotic symmetry.

Diagnostics and stop rules

Use this sequence before claiming any link in the chain.

1. Specify the boundary. Is it a finite wall, $\mathscr I^+$, $\mathscr I^-$, timelike infinity, a horizon, or a corner? List every open flux channel.
2. Specify the phase space. State dimension, topology, falloffs, endpoint conditions, allowed variations, magnetic sector, and boundary completion of the symplectic form.
3. Classify the transformation. Check that $\epsilon$ preserves the phase space and that its surface one-form is finite and globally integrable. Stop at a null direction or an obstruction.
4. Fix transport and matching. Hold a field-independent label fixed, or use the adjusted variation for a field-dependent family. State how past and future labels and Coulombic data are related.
5. Prove the complete balance. Retain matter, radiation, horizon, timelike-infinity, boundary, and corner contributions with one orientation.
6. Declare the quantum domain. Identify the states, regulator or dressing, scattering observable, anomaly status, and order of limits. Without these, stop at the classical charge law.
7. Identify the zero mode. Give its normalization and show how the soft operator acts on the chosen hard states. Without this, stop at the Ward identity.
8. Define the memory measurement. Establish endpoint convergence and a detector response. Without this, call the result an integrated radiative mode, not an observed memory.

Common failure modes

Residual is treated as physical. Solving a gauge condition only produces a residual parameter. Its admissibility, boundary pairing, and integrability still decide whether it is null or charged.

Flux is renamed breaking. A nonzero complete flux gives a balance law and can coexist with an exact symmetry. Broken symmetry, anomaly, and charge exchange are different statements.

Antipodal matching is hidden. Local field equations near $\mathscr I^+$ do not determine how data cross spatial infinity. State and justify the matching condition.

The soft limit is taken on bare charged Fock states. In four-dimensional QED that matrix element is generally not an infrared-finite observable. The state prescription and limit order are part of the claim.

Memory is identified with any charge change. Memory is an operational response to integrated radiative data under endpoint hypotheses. Matter leakage or a changing Coulombic field can alter a charge without producing the stated detector observable.

The leading Abelian result is exported unchanged. Dimension, non-Abelian covariance, confinement, gravity, subleading orders, and loops each require a new derivation.

Check your understanding

These checks have no registered assessment or completion status.

1. Isolate the soft insertion

Insert $Q^\pm=Q^{\pm,\mathrm H}+Q^{\pm,\mathrm S}$ into

$$\langle\mathrm{out}|(Q^+\mathsf S-\mathsf S Q^-)|\mathrm{in}\rangle=0$$

and solve for the terms containing $Q^{\mathrm S}$.

Solution. Linearity gives four terms. Moving the two hard terms to the right yields

$$\begin{aligned} &\langle\mathrm{out}|( Q^{+,\mathrm S}\mathsf S -\mathsf S Q^{-,\mathrm S})|\mathrm{in}\rangle \\ &\qquad=-\langle\mathrm{out}|( Q^{+,\mathrm H}\mathsf S -\mathsf S Q^{-,\mathrm H})|\mathrm{in}\rangle. \end{aligned}$$

This algebra does not yet prove a soft theorem: the zero-frequency mode and the hard action still need their model-specific normalization.

2. Recover the outward-flux sign

Suppose $\mathcal F_\epsilon[B_{12}]>0$ and

$$Q_\epsilon[C_2]-Q_\epsilon[C_1] +\mathcal F_\epsilon[B_{12}]=0.$$

Which cut has the larger charge?

Solution. Rearrangement gives $Q_\epsilon[C_2]=Q_\epsilon[C_1]-\mathcal F_\epsilon[B_{12}]$. The later cut has the smaller charge. This agrees with the finite-boundary orientation used throughout the chapter.

3. Separate a zero mode from a memory measurement

In radial gauge, assume $F_{uA}^{(0)}=\partial_uA_A^{(0)}$ and finite endpoint limits. Evaluate the time integral and identify what remains to be specified before calling it a measured memory.

Solution. The fundamental theorem of calculus gives

$$\int_{u_1}^{u_2}\!\mathrm du\,F_{uA}^{(0)} =A_A^{(0)}(u_2)-A_A^{(0)}(u_1).$$

This is an integrated radiative mode. A memory claim additionally needs a detector coupling, its initial conditions, a gauge-invariant response, and control of Coulombic fields and late-time tails.

4. Reject a divergent residual transformation

A solution of a gauge-preserving residual equation has $\epsilon_R\sim R$ on a large cut. The normal electric field scales as $R^{-2}$ and the area as $R^2$. Can this residual transformation be retained with the unrenormalized flux charge?

Solution. Its candidate charge scales as

$$Q_{\epsilon_R} \sim R\,R^{-2}R^2 \sim R,$$

so it diverges. Solving the residual gauge equation did not establish an admissible asymptotic symmetry. One must strengthen the parameter falloff or derive a physically justified renormalized phase space and repeat the integrability and flux analysis.

Exact continuations

• Soft Theorems derives the model-specific leading factors and their Abelian, non-Abelian, and gravitational orientations.
• Dressed States and Infrared-Finite Scattering develops the state-space and infrared prescriptions suppressed in the formal Ward equation above.
• Null-Infinity Radiative Data and Asymptotic Observables defines the radiative data used by the developed gravitational treatment.
• BMS, Memory, and Soft Sectors as Holographic Data develops the detailed BMS charge algebra, gravitational memory, hard–soft split, and celestial interpretation.
• Soft Limits as On-Shell Constraints treats soft behavior directly in amplitude kinematics.
• Soft Limits and Inflationary Consistency Relations is a cosmological squeezed-limit comparison. It is not a substitute for the asymptotically flat BMS and memory analysis.

References

• Araujo-Regado, Gonçalo, Philipp A. Höhn, Francesco Sartini, and Bilyana Tomova. “Soft Edges: The Many Links Between Soft and Edge Modes.” Journal of High Energy Physics 07 (2025): 180. DOI. Open PDF.
• Campiglia, Miguel, and Alok Laddha. “Asymptotic Symmetries of QED and Weinberg’s Soft Photon Theorem.” Journal of High Energy Physics 07 (2015): 115. DOI. Open PDF.
• Donnay, Laura. “Celestial Holography: An Asymptotic Symmetry Perspective.” Physics Reports 1073 (2024): 1–41. DOI. Open PDF v1.
• Gabai, Barak, and Amit Sever. “Large Gauge Symmetries and Asymptotic States in QED.” Journal of High Energy Physics 12 (2016): 095. DOI. Open PDF, substantially revised v2.
• He, Temple, Prahar Mitra, Achilleas P. Porfyriadis, and Andrew Strominger. “New Symmetries of Massless QED.” Journal of High Energy Physics 10 (2014): 112. DOI. Open PDF, corrected v2.
• Hirai, Hayato, and Sotaro Sugishita. “IR Finite S-Matrix by Gauge Invariant Dressed States.” Journal of High Energy Physics 02 (2021): 025. DOI. Open PDF.
• Pasterski, Sabrina. “Asymptotic Symmetries and Electromagnetic Memory.” Journal of High Energy Physics 09 (2017): 154. DOI. Open PDF.
• Pate, Monica, Ana-Maria Raclariu, and Andrew Strominger. “Color Memory.” Physical Review Letters 119 (2017): 261602. DOI. Open PDF.
• Prabhu, Kartik. “Conservation of Asymptotic Charges from Past to Future Null Infinity: Maxwell Fields.” Journal of High Energy Physics 10 (2018): 113. DOI. Open PDF v3.
• Prabhu, Kartik, and Gautam Satishchandran. “Infrared Finite Scattering Theory: Amplitudes and Soft Theorems.” Physical Review D 110 (2024): 085022. DOI. Open PDF, corrected v3.
• Prabhu, Kartik, Gautam Satishchandran, and Robert M. Wald. “Infrared Finite Scattering Theory in Quantum Field Theory and Quantum Gravity.” Physical Review D 106 (2022): 066005. DOI. Open PDF v3.
• Satishchandran, Gautam, and Robert M. Wald. “Asymptotic Behavior of Massless Fields and the Memory Effect.” Physical Review D 99 (2019): 084007. DOI. Open PDF, corrected v2.
• Strominger, Andrew. “Asymptotic Symmetries of Yang–Mills Theory.” Journal of High Energy Physics 07 (2014): 151. DOI. Open PDF.
• Wald, Robert M., and Andreas Zoupas. “A General Definition of ‘Conserved Quantities’ in General Relativity and Other Theories of Gravity.” Physical Review D 61 (2000): 084027. DOI. Open PDF.