Pith. sign in

REVIEW 3 major objections 5 minor 137 references

Supertranslations are Soft Dressings

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read BMS supertranslations are exactly the freedom to dress asymptotic massive states with zero-energy gravitons, and the dressing completes to a conserved charge that fixes the frame dependence of the impact parameter and angular momentum to…

desk verdict The BMS-as-dressing identification is well-earned and worth citing, but the all-orders impact-parameter claim leans on an uncontrolled eikonal remainder. read the letter →

arxiv 2608.07699 v1 pith:EUCLQV5O submitted 2026-08-07 hep-th gr-qc

Asaad Elkhidir , Donal O'Connell , Radu Roiban This is my paper · ORCID
classification hep-thgr-qc
keywords BMSsupertranslationssoftgravitonsasymptoticstatedressinglargegaugetransformationsimpactparametershiftangularmomentumframedependenceeikonalresummationclassicalscatteringamplitudes
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

In theories with massless mediators, the asymptotic state of a massive charged particle is not unique: its long-range Coulomb or gravitational field is a condensate of zero-energy quanta, and different choices of that condensate describe the same particle in different sectors. This paper establishes that the freedom to choose the condensate is exactly the BMS supertranslation freedom: dressing a massive particle with a coherent state of zero-energy gravitons of shape $T_p(n)$ changes the asymptotic shear by $C_{AB} = (\Omega_{AB} D^2 - 2 D_A D_B) T_p(n)$, so the dressing parameter is the supertranslation parameter. The dressing operator $Q_s[T]$ can be completed to a conserved charge $Q = Q_s + Q_h$ that commutes with the $S$-matrix, and an eikonal resummation of the dressed final state yields an all-orders relation between the impact parameter at infinity and the dressing data. The payoff is concrete: the known leading-order frame dependence of radiated angular momentum in general relativity and QED is reproduced as a kinematic effect of the dressing, and the same formula gives a new next-order prediction. A further result is a concrete geometric label for the two standard frames: Kerr-Schild data are the canonical BMS frame, while De Donder data are the intrinsic frame.

What carries the argument

The central object is the soft charge $Q_s[T]$, a coherent-state displacement operator built from zero-energy graviton ladder operators whose shape function is two angular derivatives of the supertranslation parameter $T_p(n)$; its classical expectation value produces the pure-gauge electromagnetic field and, in gravity, the Bondi shear $C_{AB} = (\Omega_{AB} D^2 - 2 D_A D_B) T_p(n)$. Because the $\omega=0$ support leaves $[S,Q_s]$ computable from leading soft factorization, the paper completes $Q_s$ by exact hard charges $Q_{h,0}$ and $Q_{h,M}$ built from mediator and matter number operators, so that $Q = Q_s + Q_h$ is conserved. The load-bearing technical step is the eikonal resummation of the $S$-matrix combined with a saddle-point evaluation of the dressed final state, which turns the hard-charge eigenvalue into the all-orders shift of the asymptotic impact parameter. The operator $\Omega_{AB} D^2 - 2 D_A D_B$ annihilates the $Y_{00}$ and $Y_{1m}$ harmonics of $T_p(n)$, which is why observables are independent of the monopolar and dipolar parts of the supertranslation parameter to all orders.

What would settle it

Evaluate the omitted quantum remainder $\Delta(x;s)$ in equation (5.11) at the order where $Q_h$ first acts and check whether it changes $b_{12}^2$ in equation (5.25); alternatively, compute the radiated angular-momentum difference between the canonical and intrinsic frames at next-to-leading post-Minkowskian order by an independent method and compare it with equation (6.45).

Watch

Extended reading notes

Core claim

The paper's own claim is that BMS supertranslations are not an external symmetry acting on otherwise fixed scattering data but are the same thing as the dressing ambiguity of asymptotic states: without a mass gap there is no preferred split between a massive particle and the same particle accompanied by zero-energy gravitons, and the choice of split is precisely a supertranslation. Concretely, for any function $T_p(n)$ the Hermitian soft charge $Q_s[T]$ built from zero-frequency graviton creation and annihilation operators, with shape $f_p^\eta(n) = -2\,\bar\epsilon^\eta_\mu \bar\epsilon^\eta_\nu\, \partial_n^\mu \partial_n^\nu T_p(n)$, has a classical expectation value that changes the Bondi shear by $C_{AB} = (\Omega_{AB} D^2 - 2 D_A D_B) T_p(n)$. This $Q_s$ does not by itself commute with the $S$-matrix, but its commutator is cancelled by an exact hard charge $Q_h$ computed in closed form as a functional of $T_p(n)$, so $Q = Q_s + Q_h$ satisfies $[S,Q]=0$ and acts on scattering data as the supertranslation, with the $\ell=0$ and $\ell=1$ parts of $T_p(n)$ projected out. Evaluating the dressed eikonal final state gives the all-orders impact-parameter relation $b_{12}^2 = x_\perp^2 \cos^2(\Psi/2)$ plus terms proportional to derivatives of $Q_h$; at leading order the implied mechanical angular-momentum change equals minus the known frame dependence of radiated angular momentum in general relativity and QED, and the next-order term is a new prediction. The same construction works for a scalar mediator without gauge symmetry, which the paper reads as evidence that soft factorization alone generates the conservation law.

Load-bearing premise

The all-orders impact-parameter relation assumes the classical eikonal identity (5.11) is exact, with the quantum remainder $\Delta(x;s)$ discarded; if that remainder contributes at the same order as the hard-charge terms, the all-orders shift and the next-order angular-momentum prediction would fail.

Editorial extensions

If this is right

  • Choosing a BMS frame is equivalent to choosing the coherent-state dressing of the asymptotic massive states with zero-energy gravitons, so the dressing parameter is the supertranslation parameter.
  • Observables are independent of the monopolar and dipolar parts of the supertranslation parameter to all orders, because the shear operator $\Omega_{AB} D^2 - 2 D_A D_B$ projects them out of the dressing.
  • The impact parameter measured at infinity shifts under a change of supertranslation frame, and equation (5.25) gives this shift to all orders in perturbation theory, with the entire perturbative content carried by the $S$-matrix and the closed-form hard-charge integrals.
  • At leading order the mechanical angular-momentum change implied by the shift reproduces the known frame dependence of radiated angular momentum in general relativity and QED, so total angular momentum stays frame independent, while equation (6.45) is a new next-order prediction for that frame dependence.
  • The same conserved-charge completion works for a scalar mediator with only trilinear couplings and no gauge symmetry, indicating that soft factorization alone produces the asymptotic conservation law.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the dressing equivalence is exact, the supertranslation freedom is not an ambiguity to be removed by convention but part of the definition of the scattering data; waveform comparisons between formalisms should state which zero-energy dressing, hence which BMS frame, is being used.
  • A direct test of the all-orders claim would be to compute the quantum remainder in the eikonal identity at the order where $Q_h$ first acts and see whether it changes the impact-parameter relation; if it does, the next-order angular-momentum prediction would acquire a correction proportional to that remainder.
  • The same construction should generate analogous conserved charges and frame-dependent impact parameters in mediator theories of other spins or in dimensions other than four, which would test whether the soft-factorization mechanism is as universal as the paper suggests.
  • The identification of the canonical frame with Kerr-Schild data suggests organizing classical perturbation theory directly around Kerr-Schild data, which might avoid supertranslation bookkeeping at higher post-Minkowskian orders; the authors raise this possibility but do not develop it.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper argues that BMS supertranslations in gravity, and their analogues in abelian gauge theory and in a scalar theory with no local symmetry, are equivalent to dressing massive asymptotic states with coherent states of zero-energy mediators. The dressing operator Q_s[T] is shown to produce the expected large gauge/BMS transformation of the asymptotic field, and is completed into a conserved charge Q = Q_s + Q_h by demanding [S, Q] = 0. Using eikonal methods, the paper derives an all-orders relation between the impact parameter at infinity and the eikonal impact parameter, Eq. (5.25), and applies it to the canonical-to-intrinsic BMS supertranslation. The leading-order impact-parameter shifts reproduce known results in QED and general relativity, and the next-order expression (6.45) is put forward as a new prediction for the frame dependence of radiated angular momentum.

Significance. The core identification between BMS supertranslations and soft coherent-state dressings is clean and well supported: the dressing expectation value is a pure gauge term (2.20), the gravitational shear matches the BMS transformation (3.10), and the hard charge is constructed from the soft-factorization structure rather than assumed from a local symmetry. The appendix provides closed-form evaluations of the master integrals with a stated regulator prescription, and the leading-order comparisons with Refs. [101,102] are a useful consistency check. The scalar-field example is a particularly strong illustration that the construction does not require an underlying local symmetry. If the all-orders impact-parameter relation and the next-order angular-momentum prediction are fully justified, they would constitute a substantive quantitative advance; at present that part of the paper rests on an unquantified approximation and needs further work.

major comments (3)
  1. [Sec. 5, Eq. (5.11)] The all-orders impact-parameter relation (5.25) and the next-order prediction (6.45) rest on the eikonal identity (5.11), where the paper explicitly discards a quantum remainder Delta(x;s) multiplying the phase. The magnitude of Delta is never estimated. Since the hard-charge contribution used in (6.45) is O(kappa^2 q^2) ~ O(G^3), and (6.45) is also O(G^3), any classical component of Delta at this order would modify the prediction. Please compute or bound Delta, for example from subleading soft terms or from off-shell corrections to the 2-to-2 amplitude, or else state explicitly that (5.25) and (6.45) are valid only under the assumption Delta = 0.
  2. [Sec. 5, Eqs. (5.13)-(5.25)] The derivation of Eq. (5.25) also uses a saddle-point evaluation of the x and q integrals and identifies q = q_* with the impulse. The paper calls the resulting relation exact and valid to all orders, but the status of the saddle-point approximation at arbitrary post-Minkowskian order is not discussed. In the classical limit the stationary-phase approximation is standard, but it introduces corrections of the same type as the discarded Delta(x;s). The authors should qualify the all-orders claim as holding in the classical eikonal approximation and demonstrate that the saddle point captures the hard-charge derivative terms at the order used in (6.45).
  3. [Sec. 6.1, Eqs. (6.6) and (6.16)] The abstract and Sec. 5 state that observables are independent of the monopolar and dipolar parts of the BMS parameter to all orders. The construction in Sec. 4 projects out ell=0,1 modes from the hard charge, but Sec. 6.1 deliberately keeps them by working in a convenient 'supertranslation gauge'. The paper does not explicitly show that the final physical impact-parameter shift is unchanged by this choice. Because the leading-order comparison only checks the first post-Minkowskian order, the all-orders cancellation should be demonstrated or explicitly restricted to the projected construction.
minor comments (5)
  1. [Sec. 2.1, Eq. (2.4)] The phrase 'all possible invariant masses s 1/2' appears garbled and should read 'sqrt(s)' or be reworded.
  2. [Appendix B, Eqs. (B.2)-(B.4)] The notation uses K for both the four-vector and its spatial norm, which is confusing in the massless limit K -> 1; a different symbol for the norm would improve readability.
  3. [Figures 1 and 2] The captions refer to a 'red wiggly line' but the printed figures may be monochrome; labeling the line types explicitly would avoid ambiguity.
  4. [Sec. 5, Eq. (5.27)] The notation O(g^2) and O(g^4) for the coupling expansion is used without defining the normalization of g; please state the convention once.
  5. [Sec. 6.2] The footnotes explaining that the expansion in q is the PM expansion are helpful, but the distinction between the integration variable q, the impulse, and the momentum transfer could be stated more prominently in the main text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the dressing-to-BMS map and the all-orders impact-parameter relation are derived from explicit commutator and eikonal computations, not from fitted inputs or self-citations.

full rationale

The central identification is a direct calculation: Eq. (3.10) computes the shear of the dressed state as C_AB = (Omega_AB D^2 - 2 D_A D_B) T_p(n), which is the standard BMS supertranslation shear, and Eq. (2.20) gives the pure-gauge vector analog. The hard charge is not fitted: Q_h is fixed by the requirement [S, Q_s + Q_h] = 0 in Sec. 4, with the master integrals evaluated in Appendix B, and its eigenvalue on a two-particle state is then used in the eikonal saddle-point derivation of Eq. (5.25). The leading-order impact-parameter shifts in Eqs. (6.43) and (6.44) are checked against the independent external results of Refs. [101] and [102], while the O(G^3) expression (6.45) is a genuinely new prediction with no free parameter adjusted to produce it. The paper's reliance on the authors' earlier Ref. [91] is framing: it concerns the interpretation of zero-energy three-point elements and the separation of soft content, not the derivation of the central formulas, which are self-contained in this paper. The discarded quantum remainder in Eq. (5.11) is an explicitly stated approximation, a correctness risk rather than a circular input, because the paper does not use the remainder to define the result it then 'predicts'.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

No new particles, fields, or entities are postulated. The soft and hard charges are composite operators built from existing fields. The derivation relies on standard QFT axioms, the idealization of strictly zero-energy dressings, the classical eikonal resummation, and a specific regulator prescription; these are the main sources of potential fragility.

free parameters (1)
  • Supertranslation parameter T_p(n) (zero-energy dressing shape)
    Arbitrary real function on the celestial sphere defining the coherent-state dressing and hence the asymptotic frame (Eq. 2.8, Eq. 2.12). It is the large gauge/BMS parameter under study, not fitted to data; the paper argues observables are independent of its ell=0,1 modes. The specific canonical-to-intrinsic choice (Eq. 6.1) is fixed by the Kerr-Schild to de Donder diffeomorphism.
assumptions (7)
  • domain assumption Standard QFT axioms (Poincare invariance, positivity/unitarity, vacuum, physical spectrum) for the Kallan-Lehmann representation
    Invoked in Sec. 2.1 to establish the spectral representation and the infraparticle structure of charged fields.
  • domain assumption Stationary phase approximation for asymptotic fields and its extension to zero frequency
    Eqs. (1.6)-(1.8); the field at large distance is defined by the stationary phase expression even for small or zero frequency, which is an idealization noted by the authors.
  • domain assumption Zero-energy dressing idealization: shape function F_p^eta(k) = delta(omega) f_p^eta(n), with the cutoff much smaller than any other scale
    Eq. (2.8) and Sec. 2.2; the strict omega = 0 limit is a theoretical construct, with the limit taken after all commutators are evaluated.
  • domain assumption Classical eikonal resummation with quantum corrections discarded
    Eq. (5.11); the paper states 'we are implicitly discarding quantum corrections' and does not control the remainder Delta(x;s).
  • domain assumption Static (u-independent) dressing in the infinite past
    Sec. 2.2 and Conclusions; the dressing captures the asymptotic field of initial-state particles located strictly in the infinite past, with time-dependent extensions left to future work.
  • domain assumption Bondi coordinates and peeling property hold for the scattering spacetimes considered, at the perturbative level
    Sec. 3.2, footnote 13; the paper notes that Ref. [120] raises questions about Bondi-coordinate existence with IR dressings, but treats spacetime perturbatively.
  • ad hoc to paper The beta-regulator with the order of limits d to 4 first, then beta to 0, yields unambiguous hard charges
    Introduced in Sec. 4.1 (Eq. 4.8) to move the support of the soft integral away from the origin; the paper asserts but does not prove regulator independence.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Supertranslations are Soft Dressings." pith.science (2026). https://pith.science/paper/EUCLQV5O

@misc{pith2026260807699,
  author       = {Pith},
  title        = {Pith review of: Supertranslations are Soft Dressings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EUCLQV5O}},
  note         = {Machine review of arXiv:2608.07699}
}
read the original abstract

Asymptotic states of massive particles in electrodynamics and gravity are not uniquely defined because of the absence of a mass gap. Working in the KMOC formalism, we study the classical implications of this freedom in the dressing of massive particle states. We show that coherent-state dressings modify observables by a large gauge/BMS transformation. The associated coherent-state displacement operator can be completed into a conserved charge, the BMS charge in the gravitational case, which implements large gauge transformations on scattering data. The parameter of our dressing is directly the large gauge/BMS transformation parameter, and we explain that observables do not depend on monopolar and dipolar parts of the transformation parameter to all orders. Our construction is driven by soft factorization rather than by an underlying local symmetry, and we illustrate this by exhibiting the same structure in a scalar theory with no gauge symmetry. We explore the action of large gauge transformations on two-particle states for scalar, abelian gauge and graviton mediators, obtaining a universal shift of the impact parameter and clarifying the frame dependence of the radiated angular momentum vis \`a vis the frame-independence of the total angular momentum. The transformation connecting the canonical BMS frame, in which the Schwarzschild metric takes the Kerr-Schild form, to the intrinsic frame, in which it takes the De Donder form, illustrates these effects concretely.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

137 extracted references · 10 canonical work pages

  1. [3]

    Punturo et al.,The Einstein Telescope: A third-generation gravitational wave observatory,Class

    M. Punturo et al.,The Einstein Telescope: A third-generation gravitational wave observatory,Class. Quant. Grav.27(2010) 194002. [4]LISAcollaboration,Laser Interferometer Space Antenna,1702.00786

    arXiv 2010

  2. [5]

    Reitze et al.,Cosmic Explorer: The U.S

    D. Reitze et al.,Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO,Bull. Am. Astron. Soc.51(2019) 035 [1907.04833]

    arXiv 2019

  3. [6]

    Report from the LSC Post-O5 Study Group

    P. Fritschel et al., “Report from the LSC Post-O5 Study Group.”

  4. [7]

    Abac et al.,The Science of the Einstein Telescope,2503.12263

    A. Abac et al.,The Science of the Einstein Telescope,2503.12263

    arXiv

  5. [8]

    Pretorius,Evolution of binary black hole spacetimes,Phys

    F. Pretorius,Evolution of binary black hole spacetimes,Phys. Rev. Lett.95(2005) 121101 [gr-qc/0507014]

    arXiv 2005

  6. [9]

    Campanelli, C.O

    M. Campanelli, C.O. Lousto, P. Marronetti and Y. Zlochower,Accurate evolutions of orbiting black-hole binaries without excision,Phys. Rev. Lett.96(2006) 111101 [gr-qc/0511048]

    arXiv 2006

  7. [10]

    Baker, J

    J.G. Baker, J. Centrella, D.-I. Choi, M. Koppitz and J. van Meter,Gravitational wave extraction from an inspiraling configuration of merging black holes,Phys. Rev. Lett.96 (2006) 111102 [gr-qc/0511103]

    arXiv 2006

  8. [11]

    Damour, F

    T. Damour, F. Guercilena, I. Hinder, S. Hopper, A. Nagar and L. Rezzolla,Strong-Field Scattering of Two Black Holes: Numerics Versus Analytics,Phys. Rev. D89(2014) 081503 [1402.7307]

    arXiv 2014

Show all 137 references
  1. [12]

    Y. Mino, M. Sasaki and T. Tanaka,Gravitational radiation reaction to a particle motion, Phys. Rev. D55(1997) 3457 [gr-qc/9606018]

    1997 arXiv

  2. [13]

    Quinn and R.M

    T.C. Quinn and R.M. Wald,An Axiomatic approach to electromagnetic and gravitational radiation reaction of particles in curved space-time,Phys. Rev. D56(1997) 3381 [gr-qc/9610053]

    1997 arXiv

  3. [14]

    Poisson, A

    E. Poisson, A. Pound and I. Vega,The Motion of point particles in curved spacetime,Living Rev. Rel.14(2011) 7 [1102.0529]

    2011 arXiv

  4. [15]

    Barack and A

    L. Barack and A. Pound,Self-force and radiation reaction in general relativity,Rept. Prog. Phys.82(2019) 016904 [1805.10385]

    2019 arXiv

  5. [16]

    Cheung, I.Z

    C. Cheung, I.Z. Rothstein and M.P. Solon,From Scattering Amplitudes to Classical Potentials in the Post-Minkowskian Expansion,Phys. Rev. Lett.121(2018) 251101 [1808.02489]

    2018 arXiv

  6. [17]

    Di Vecchia, C

    P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano,Universality of ultra-relativistic gravitational scattering,Phys. Lett. B811(2020) 135924 [2008.12743]

    2020 arXiv

  7. [18]

    Di Vecchia, C

    P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano,The eikonal approach to gravitational scattering and radiation atO(G 3),JHEP07(2021) 169 [2104.03256]. – 51 –

    2021 arXiv

  8. [19]

    Damgaard, L

    P.H. Damgaard, L. Plante and P. Vanhove,On an exponential representation of the gravitational S-matrix,JHEP11(2021) 213 [2107.12891]

    2021 arXiv

  9. [20]

    Damgaard, E.R

    P.H. Damgaard, E.R. Hansen, L. Plant´ e and P. Vanhove,Classical observables from the exponential representation of the gravitational S-matrix,JHEP09(2023) 183 [2307.04746]

    2023 arXiv

  10. [21]

    Kosower, B

    D.A. Kosower, B. Maybee and D. O’Connell,Amplitudes, Observables, and Classical Scattering,JHEP02(2019) 137 [1811.10950]

    2019 arXiv

  11. [22]

    Damgaard, K

    P.H. Damgaard, K. Haddad and A. Helset,Heavy Black Hole Effective Theory,JHEP11 (2019) 070 [1908.10308]

    2019 arXiv

  12. [23]

    Cheung, J

    C. Cheung, J. Parra-Martinez, I.Z. Rothstein, N. Shah and J. Wilson-Gerow,Effective Field Theory for Extreme Mass Ratio Binaries,Phys. Rev. Lett.132(2024) 091402 [2308.14832]

    2024 arXiv

  13. [24]

    Kosmopoulos and M.P

    D. Kosmopoulos and M.P. Solon,Gravitational self force from scattering amplitudes in curved space,JHEP03(2024) 125 [2308.15304]

    2024 arXiv

  14. [25]

    K¨ alin, Z

    G. K¨ alin, Z. Liu and R.A. Porto,Conservative Dynamics of Binary Systems to Third Post-Minkowskian Order from the Effective Field Theory Approach,Phys. Rev. Lett.125 (2020) 261103 [2007.04977]

    2020 arXiv

  15. [26]

    Mogull, J

    G. Mogull, J. Plefka and J. Steinhoff,Classical black hole scattering from a worldline quantum field theory,JHEP02(2021) 048 [2010.02865]

    2021 arXiv

  16. [27]

    K¨ alin, J

    G. K¨ alin, J. Neef and R.A. Porto,Radiation-reaction in the Effective Field Theory approach to Post-Minkowskian dynamics,JHEP01(2023) 140 [2207.00580]

    2023 arXiv

  17. [28]

    Jakobsen, G

    G.U. Jakobsen, G. Mogull, J. Plefka and B. Sauer,All things retarded: radiation-reaction in worldline quantum field theory,JHEP10(2022) 128 [2207.00569]

    2022 arXiv

  18. [29]

    Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M.P. Solon and M. Zeng,Black Hole Binary Dynamics from the Double Copy and Effective Theory,JHEP10(2019) 206 [1908.01493]

    2019 arXiv

  19. [30]

    Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M.P. Solon and M. Zeng,Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order,Phys. Rev. Lett.122(2019) 201603 [1901.04424]

    2019 arXiv

  20. [31]

    Z. Bern, E. Herrmann, R. Roiban, M.S. Ruf, A.V. Smirnov, V.A. Smirnov et al., Amplitudes, supersymmetric black hole scattering atO G5 , and loop integration,JHEP10 (2024) 023 [2406.01554]

    2024

  21. [32]

    Z. Bern, J. Parra-Martinez, R. Roiban, M.S. Ruf, C.-H. Shen, M.P. Solon et al.,Scattering Amplitudes and Conservative Binary Dynamics atO(G 4),Phys. Rev. Lett.126(2021) 171601 [2101.07254]

    2021 arXiv

  22. [33]

    Z. Bern, J. Parra-Martinez, R. Roiban, M.S. Ruf, C.-H. Shen, M.P. Solon et al.,Scattering Amplitudes, the Tail Effect, and Conservative Binary Dynamics atO(G 4),Phys. Rev. Lett. 128(2022) 161103 [2112.10750]

    2022 arXiv

  23. [34]

    Driesse, G.U

    M. Driesse, G.U. Jakobsen, G. Mogull, J. Plefka, B. Sauer and J. Usovitsch,Conservative Black Hole Scattering at Fifth Post-Minkowskian and First Self-Force Order,Phys. Rev. Lett.132(2024) 241402 [2403.07781]

    2024 arXiv

  24. [35]

    Driesse, G.U

    M. Driesse, G.U. Jakobsen, A. Klemm, G. Mogull, C. Nega, J. Plefka et al.,Emergence of Calabi–Yau manifolds in high-precision black-hole scattering,Nature641(2025) 603 [2411.11846]. – 52 –

    2025 arXiv

  25. [36]

    Jakobsen, G

    G.U. Jakobsen, G. Mogull, J. Plefka and B. Sauer,Dissipative Scattering of Spinning Black Holes at Fourth Post-Minkowskian Order,Phys. Rev. Lett.131(2023) 241402 [2308.11514]

    2023 arXiv

  26. [37]

    Jakobsen, G

    G.U. Jakobsen, G. Mogull, J. Plefka, B. Sauer and Y. Xu,Conservative Scattering of Spinning Black Holes at Fourth Post-Minkowskian Order,Phys. Rev. Lett.131(2023) 151401 [2306.01714]

    2023 arXiv

  27. [38]

    Dlapa, G

    C. Dlapa, G. K¨ alin, Z. Liu and R.A. Porto,Conservative Dynamics of Binary Systems at Fourth Post-Minkowskian Order in the Large-Eccentricity Expansion,Phys. Rev. Lett.128 (2022) 161104 [2112.11296]

    2022 arXiv

  28. [39]

    Dlapa, G

    C. Dlapa, G. K¨ alin, Z. Liu and R.A. Porto,Dynamics of binary systems to fourth Post-Minkowskian order from the effective field theory approach,Phys. Lett. B831(2022) 137203 [2106.08276]

    2022 arXiv

  29. [40]

    Bjerrum-Bohr, L

    N.E.J. Bjerrum-Bohr, L. Plant´ e and P. Vanhove,Post-Minkowskian radial action from soft limits and velocity cuts,JHEP03(2022) 071 [2111.02976]

    2022 arXiv

  30. [41]

    Z. Bern, E. Herrmann, R. Roiban, M.S. Ruf, A.V. Smirnov, S. Smith et al.,Scattering Amplitudes and Conservative Binary Dynamics atO(G 5)without Self-Force Truncation, 2512.23654

  31. [42]

    Driesse, G.U

    M. Driesse, G.U. Jakobsen, G. Mogull, C. Nega, J. Plefka, B. Sauer et al.,Conservative Black Hole Scattering at Fifth Post-Minkowskian and Second Self-Force Order,2601.16256

    arXiv

  32. [43]

    Herderschee, R

    A. Herderschee, R. Roiban and F. Teng,The sub-leading scattering waveform from amplitudes,JHEP06(2023) 004 [2303.06112]

    2023 arXiv

  33. [44]

    Elkhidir, D

    A. Elkhidir, D. O’Connell, M. Sergola and I.A. Vazquez-Holm,Radiation and Reaction at One Loop,2303.06211

    arXiv

  34. [45]

    Georgoudis, C

    A. Georgoudis, C. Heissenberg and R. Russo,An eikonal-inspired approach to the gravitational scattering waveform,JHEP03(2024) 089 [2312.07452]

    2024 arXiv

  35. [46]

    Georgoudis, C

    A. Georgoudis, C. Heissenberg and I. Vazquez-Holm,Inelastic exponentiation and classical gravitational scattering at one loop,JHEP2023(2023) 126 [2303.07006]

    2023 arXiv

  36. [47]

    Brandhuber, G.R

    A. Brandhuber, G.R. Brown, G. Chen, S. De Angelis, J. Gowdy and G. Travaglini, One-loop gravitational bremsstrahlung and waveforms from a heavy-mass effective field theory,JHEP06(2023) 048 [2303.06111]

    2023 arXiv

  37. [48]

    D. Bini, T. Damour, S. De Angelis, A. Geralico, A. Herderschee, R. Roiban et al., Gravitational Waveform: A Tale of Two Formalisms,2402.06604

    arXiv

  38. [49]

    Blanchet,Post-Newtonian Theory for Gravitational Waves,Living Rev

    L. Blanchet,Post-Newtonian Theory for Gravitational Waves,Living Rev. Rel.17(2014) 2 [1310.1528]

    2014 arXiv

  39. [50]

    Porto,The effective field theorist’s approach to gravitational dynamics,Phys

    R.A. Porto,The effective field theorist’s approach to gravitational dynamics,Phys. Rept. 633(2016) 1 [1601.04914]

    2016 arXiv

  40. [51]

    Buonanno, M

    A. Buonanno, M. Khalil, D. O’Connell, R. Roiban, M.P. Solon and M. Zeng,Snowmass White Paper: Gravitational Waves and Scattering Amplitudes, inSnowmass 2021, 4, 2022 [2204.05194]

    2021 arXiv

  41. [52]

    Barausse, V

    E. Barausse, V. Cardoso and P. Pani,Can environmental effects spoil precision gravitational-wave astrophysics?,Phys. Rev. D89(2014) 104059 [1404.7149]

    2014 arXiv

  42. [53]

    Torre,Gravitational observables and local symmetries,Phys

    C.G. Torre,Gravitational observables and local symmetries,Phys. Rev. D48(1993) R2373 [gr-qc/9306030]. – 53 –

    1993 arXiv

  43. [54]

    Giddings, D

    S.B. Giddings, D. Marolf and J.B. Hartle,Observables in effective gravity,Phys. Rev. D74 (2006) 064018 [hep-th/0512200]

    2006 arXiv

  44. [55]

    DeWitt,Quantum Theory of Gravity

    B.S. DeWitt,Quantum Theory of Gravity. 1. The Canonical Theory,Phys. Rev.160(1967) 1113

    1967

  45. [56]

    Donnelly and S.B

    W. Donnelly and S.B. Giddings,Diffeomorphism-invariant observables and their nonlocal algebra,Phys. Rev. D93(2016) 024030 [1507.07921]

    2016 arXiv

  46. [57]

    Bergmann,Observables in General Relativity,Rev

    P.G. Bergmann,Observables in General Relativity,Rev. Mod. Phys.33(1961) 510

    1961

  47. [58]

    Cheung, A

    C. Cheung, A. Sivaramakrishnan, J. Wilson-Gerow and L. Zhou,On Perturbatively Dressed Observables,2605.26077

    arXiv

  48. [59]

    Dirac,Gauge invariant formulation of quantum electrodynamics,Can

    P.A.M. Dirac,Gauge invariant formulation of quantum electrodynamics,Can. J. Phys.33 (1955) 650

    1955

  49. [60]

    Mandelstam,Quantum electrodynamics without potentials,Annals Phys.19(1962) 1

    S. Mandelstam,Quantum electrodynamics without potentials,Annals Phys.19(1962) 1

    1962

  50. [61]

    Lavelle and D

    M. Lavelle and D. McMullan,Constituent quarks from QCD,Phys. Rept.279(1997) 1 [hep-ph/9509344]

    1997 arXiv

  51. [62]

    Veneziano and G.A

    G. Veneziano and G.A. Vilkovisky,Angular momentum loss in gravitational scattering, radiation reaction, and the Bondi gauge ambiguity,Phys. Lett. B834(2022) 137419 [2201.11607]

    2022 arXiv

  52. [63]

    Barnich and C

    G. Barnich and C. Troessaert,Finite BMS transformations,JHEP03(2016) 167 [1601.04090]

    2016 arXiv

  53. [64]

    Strominger,On BMS Invariance of Gravitational Scattering,JHEP07(2014) 152 [1312.2229]

    A. Strominger,On BMS Invariance of Gravitational Scattering,JHEP07(2014) 152 [1312.2229]

    2014 arXiv

  54. [65]

    Strominger and A

    A. Strominger and A. Zhiboedov,Gravitational Memory, BMS Supertranslations and Soft Theorems,JHEP01(2016) 086 [1411.5745]

    2016 arXiv

  55. [66]

    Campiglia and A

    M. Campiglia and A. Laddha,Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev. D90(2014) 124028 [1408.2228]

    2014 arXiv

  56. [67]

    T. He, V. Lysov, P. Mitra and A. Strominger,BMS supertranslations and Weinberg’s soft graviton theorem,JHEP05(2015) 151 [1401.7026]

    2015 arXiv

  57. [68]

    Pasterski, A

    S. Pasterski, A. Strominger and A. Zhiboedov,New Gravitational Memories,JHEP12 (2016) 053 [1502.06120]

    2016 arXiv

  58. [69]

    Campiglia and A

    M. Campiglia and A. Laddha,New symmetries for the Gravitational S-matrix,JHEP04 (2015) 076 [1502.02318]

    2015 arXiv

  59. [70]

    Campiglia and A

    M. Campiglia and A. Laddha,Asymptotic symmetries of QED and Weinberg’s soft photon theorem,JHEP07(2015) 115 [1505.05346]

    2015 arXiv

  60. [71]

    Conde and P

    E. Conde and P. Mao,Remarks on asymptotic symmetries and the subleading soft photon theorem,Phys. Rev. D95(2017) 021701 [1605.09731]

    2017 arXiv

  61. [72]

    Kapec, V

    D. Kapec, V. Lysov, S. Pasterski and A. Strominger,Higher-dimensional supertranslations and Weinberg’s soft graviton theorem,Ann. Math. Sci. Appl.02(2017) 69 [1502.07644]

    2017 arXiv

  62. [73]

    Henneaux and C

    M. Henneaux and C. Troessaert,BMS Group at Spatial Infinity: the Hamiltonian (ADM) approach,JHEP03(2018) 147 [1801.03718]

    2018 arXiv

  63. [74]

    Kapec, M

    D. Kapec, M. Perry, A.-M. Raclariu and A. Strominger,Infrared Divergences in QED, Revisited,Phys. Rev. D96(2017) 085002 [1705.04311]. – 54 –

    2017 arXiv

  64. [75]

    Himwich, S.A

    E. Himwich, S.A. Narayanan, M. Pate, N. Paul and A. Strominger,The SoftS-Matrix in Gravity,JHEP09(2020) 129 [2005.13433]

    2020 arXiv

  65. [76]

    Campiglia and A

    M. Campiglia and A. Laddha,Asymptotic symmetries of gravity and soft theorems for massive particles,JHEP12(2015) 094 [1509.01406]

    2015 arXiv

  66. [77]

    Strominger,Lectures on the Infrared Structure of Gravity and Gauge Theory(3, 2017), [1703.05448]

    A. Strominger,Lectures on the Infrared Structure of Gravity and Gauge Theory(3, 2017), [1703.05448]

    2017 arXiv

  67. [78]

    Ashtekar, M

    A. Ashtekar, M. Campiglia and A. Laddha,Null infinity, the BMS group and infrared issues,Gen. Rel. Grav.50(2018) 140 [1808.07093]

    2018 arXiv

  68. [79]

    Raclariu,Lectures on Celestial Holography,2107.02075

    A.-M. Raclariu,Lectures on Celestial Holography,2107.02075

    arXiv

  69. [80]

    Pasterski,Lectures on celestial amplitudes,Eur

    S. Pasterski,Lectures on celestial amplitudes,Eur. Phys. J. C81(2021) 1062 [2108.04801]

    2021 arXiv

  70. [81]

    Mirbabayi and M

    M. Mirbabayi and M. Porrati,Dressed Hard States and Black Hole Soft Hair,Phys. Rev. Lett.117(2016) 211301 [1607.03120]

    2016 arXiv

  71. [82]

    Bousso and M

    R. Bousso and M. Porrati,Soft Hair as a Soft Wig,Class. Quant. Grav.34(2017) 204001 [1706.00436]

    2017 arXiv

  72. [83]

    Flanagan and I

    E.E. Flanagan and I. Shehzad,The classical dynamics of gauge theories in the deep infrared,JHEP05(2023) 185 [2210.11585]

    2023 arXiv

  73. [84]

    Satishchandran and R.M

    G. Satishchandran and R.M. Wald,Asymptotic behavior of massless fields and the memory effect,Phys. Rev. D99(2019) 084007

    2019

  74. [85]

    Prabhu, G

    K. Prabhu, G. Satishchandran and R.M. Wald,Infrared finite scattering theory in quantum field theory and quantum gravity,Phys. Rev. D106(2022) 066005 [2203.14334]

    2022 arXiv

  75. [86]

    Cristofoli, R

    A. Cristofoli, R. Gonzo, D.A. Kosower and D. O’Connell,Waveforms from amplitudes, Phys. Rev. D106(2022) 056007 [2107.10193]

    2022 arXiv

  76. [87]

    Carney, L

    D. Carney, L. Chaurette, D. Neuenfeld and G. Semenoff,On the need for soft dressing, JHEP09(2018) 121 [1803.02370]

    2018 arXiv

  77. [88]

    Lippstreu,Analytic Properties of Infrared-Finite Amplitudes in Theories with Long-Range Forces,2505.04702

    L. Lippstreu,Analytic Properties of Infrared-Finite Amplitudes in Theories with Long-Range Forces,2505.04702

    arXiv

  78. [89]

    Chicherin, G.P

    D. Chicherin, G.P. Korchemsky, E. Sokatchev and A. Zhiboedov,Energy correlators in four-dimensional gravity,2512.23791

    arXiv

  79. [90]

    Caron-Huot, M

    S. Caron-Huot, M. Giroux, H.S. Hannesdottir and S. Mizera,What can be measured asymptotically?,2308.02125

    arXiv

  80. [91]

    Elkhidir, D

    A. Elkhidir, D. O’Connell and R. Roiban,Supertranslations from Scattering Amplitudes, Phys. Rev. Lett.135(2025) 151601 [2408.15961]

    2025 arXiv

  81. [92]

    Blanchet and T

    L. Blanchet and T. Damour,Radiative gravitational fields in general relativity I. general structure of the field outside the source,Phil. Trans. Roy. Soc. Lond. A320(1986) 379

    1986

  82. [93]

    Blanchet and T

    L. Blanchet and T. Damour,Postnewtonian Generation of Gravitational Waves,Ann. Inst. H. Poincare Phys. Theor.50(1989) 377

    1989

  83. [94]

    Bonga and E

    B. Bonga and E. Poisson,Coulombic contribution to angular momentum flux in general relativity,Phys. Rev. D99(2019) 064024 [1808.01288]

    2019 arXiv

  84. [95]

    Menezes,w 1+∞ as the Frame Algebra of Kerr Soft Dressing,2607.27469

    G. Menezes,w 1+∞ as the Frame Algebra of Kerr Soft Dressing,2607.27469

    arXiv

  85. [96]

    Menezes,Kerr Soft Dressing and thew 1+∞ Frame Algebra at Null Infinity,2607.27478

    G. Menezes,Kerr Soft Dressing and thew 1+∞ Frame Algebra at Null Infinity,2607.27478. – 55 –

    arXiv

  86. [97]

    S. Choi, S. Sandeep Pradhan and R. Akhoury,Supertranslation Hair of Schwarzschild Black Hole: A Wilson Line Perspective,JHEP01(2020) 013 [1910.05882]

    2020 arXiv

  87. [98]

    Choi and R

    S. Choi and R. Akhoury,BMS Supertranslation Symmetry Implies Faddeev-Kulish Amplitudes,JHEP02(2018) 171 [1712.04551]

    2018 arXiv

  88. [99]

    Choi and R

    S. Choi and R. Akhoury,Soft Photon Hair on Schwarzschild Horizon from a Wilson Line Perspective,JHEP12(2018) 074 [1809.03467]

    2018 arXiv

  89. [100]

    Comp` ere, S.E

    G. Comp` ere, S.E. Gralla and H. Wei,An asymptotic framework for gravitational scattering, Class. Quant. Grav.40(2023) 205018 [2303.17124]

    2023 arXiv

  90. [101]

    Bini and T

    D. Bini and T. Damour,Radiation-reaction and angular momentum loss at the second post-Minkowskian order,Phys. Rev. D106(2022) 124049 [2211.06340]

    2022 arXiv

  91. [102]

    Saketh, J

    M.V.S. Saketh, J. Vines, J. Steinhoff and A. Buonanno,Conservative and radiative dynamics in classical relativistic scattering and bound systems,Phys. Rev. Res.4(2022) 013127 [2109.05994]

    2022 arXiv

  92. [103]

    Kallen,On the definition of the Renormalization Constants in Quantum Electrodynamics,Helv

    G. Kallen,On the definition of the Renormalization Constants in Quantum Electrodynamics,Helv. Phys. Acta25(1952) 417

    1952

  93. [104]

    Lehmann,On the Properties of propagation functions and renormalization contants of quantized fields,Nuovo Cim.11(1954) 342

    H. Lehmann,On the Properties of propagation functions and renormalization contants of quantized fields,Nuovo Cim.11(1954) 342

    1954

  94. [105]

    Collins,A new approach to the LSZ reduction formula,1904.10923

    J. Collins,A new approach to the LSZ reduction formula,1904.10923

    1904 arXiv

  95. [106]

    Yennie, S.C

    D.R. Yennie, S.C. Frautschi and H. Suura,The infrared divergence phenomena and high-energy processes,Annals Phys.13(1961) 379

    1961

  96. [107]

    Weinberg,Infrared photons and gravitons,Phys

    S. Weinberg,Infrared photons and gravitons,Phys. Rev.140(1965) B516

    1965

  97. [108]

    Bloch and A

    F. Bloch and A. Nordsieck,Note on the Radiation Field of the electron,Phys. Rev.52 (1937) 54

    1937

  98. [109]

    Lee and M

    T.D. Lee and M. Nauenberg,Degenerate Systems and Mass Singularities,Phys. Rev.133 (1964) B1549

    1964

  99. [110]

    Kinoshita,Mass singularities of Feynman amplitudes,J

    T. Kinoshita,Mass singularities of Feynman amplitudes,J. Math. Phys.3(1962) 650

    1962

  100. [111]

    Schroer,Infraparticles in quantum field theory,Fortsch

    B. Schroer,Infraparticles in quantum field theory,Fortsch. Phys.11(1963) 1

    1963

  101. [112]

    Frohlich,THE CHARGED SECTORS OF QUANTUM ELECTRODYNAMICS IN A FRAMEWORK OF LOCAL OBSER V ABLES,Commun

    J. Frohlich,THE CHARGED SECTORS OF QUANTUM ELECTRODYNAMICS IN A FRAMEWORK OF LOCAL OBSER V ABLES,Commun. Math. Phys.66(1979) 223

    1979

  102. [113]

    Buchholz,Gauss’ Law and the Infraparticle Problem,Phys

    D. Buchholz,Gauss’ Law and the Infraparticle Problem,Phys. Lett. B174(1986) 331

    1986

  103. [114]

    Kulish and L.D

    P.P. Kulish and L.D. Faddeev,Asymptotic conditions and infrared divergences in quantum electrodynamics,Theor. Math. Phys.4(1970) 745

    1970

  104. [115]

    Dybalski,From Faddeev–Kulish to LSZ

    W. Dybalski,From Faddeev–Kulish to LSZ. Towards a non-perturbative description of colliding electrons,Nucl. Phys. B925(2017) 455 [1706.09057]

    2017 arXiv

  105. [116]

    Gaß, K.-H

    C. Gaß, K.-H. Rehren and F.C. Tippner,On the spacetime structure of infrared divergencies in QED,Lett. Math. Phys.112(2022) 37 [2109.10148]

    2022 arXiv

  106. [117]

    Bagan, M

    E. Bagan, M. Lavelle and D. McMullan,Charges from dressed matter: Construction, Annals Phys.282(2000) 471 [hep-ph/9909257]

    2000 arXiv

  107. [118]

    X. Feal, A. Tarasov and R. Venugopalan,QED as a many-body theory of worldlines: General formalism and infrared structure,Phys. Rev. D106(2022) 056009 [2206.04188]. – 56 –

    2022 arXiv

  108. [119]

    X. Feal, A. Tarasov and R. Venugopalan,QED as a many-body theory of worldlines. II. All-order S-matrix formalism,Phys. Rev. D107(2023) 096021 [2211.15712]

    2023 arXiv

  109. [120]

    De Angelis, A

    S. De Angelis, A. Herderschee, R. Roiban and F. Teng,Asymptotic Simplicity and Scattering in General Relativity from Quantum Field Theory,2511.10637

  110. [121]

    Newman and R

    E.T. Newman and R. Penrose,Note on the Bondi-Metzner-Sachs group,J. Math. Phys.7 (1966) 863

    1966

  111. [122]

    Goldberg, A.J

    J.N. Goldberg, A.J. MacFarlane, E.T. Newman, F. Rohrlich and E.C.G. Sudarshan,Spin-s spherical harmonics andð,J. Math. Phys.8(1967) 2155

    1967

  112. [123]

    Eastwood and K.P

    M.G. Eastwood and K.P. Tod,Edth—a differential operator on the sphere,Mathematical Proceedings of the Cambridge Philosophical Society92(1982) 317

    1982

  113. [124]

    Monteiro, D

    R. Monteiro, D. O’Connell and C.D. White,Black holes and the double copy,JHEP12 (2014) 056 [1410.0239]

    2014 arXiv

  114. [125]

    Vines,Scattering of two spinning black holes in post-Minkowskian gravity, to all orders in spin, and effective-one-body mappings,Class

    J. Vines,Scattering of two spinning black holes in post-Minkowskian gravity, to all orders in spin, and effective-one-body mappings,Class. Quant. Grav.35(2018) 084002 [1709.06016]

    2018 arXiv

  115. [126]

    Flanagan and D.A

    ´E.´E. Flanagan and D.A. Nichols,Conserved charges of the extended Bondi-Metzner-Sachs algebra,Phys. Rev. D95(2017) 044002 [1510.03386]

    2017 arXiv

  116. [127]

    D. Bini, T. Damour and A. Geralico,Quadrupolar bremsstrahlung waveform at the third-and-a-half post-Newtonian accuracy,2604.21522

    arXiv

  117. [128]

    Comp` ere, R

    G. Comp` ere, R. Oliveri and A. Seraj,The Poincar´ e and BMS flux-balance laws with application to binary systems,JHEP10(2020) 116 [1912.03164]

    2020 arXiv

  118. [129]

    Cristofoli, R

    A. Cristofoli, R. Gonzo, N. Moynihan, D. O’Connell, A. Ross, M. Sergola et al.,The Uncertainty Principle and Classical Amplitudes,2112.07556

    arXiv

  119. [130]

    Campiglia, L

    M. Campiglia, L. Freidel, F. Hopfmueller and R.M. Soni,Scalar Asymptotic Charges and Dual Large Gauge Transformations,JHEP04(2019) 003 [1810.04213]

    2019 arXiv

  120. [131]

    Campiglia and A

    M. Campiglia and A. Laddha,Asymptotic charges in massless QED revisited: A view from Spatial Infinity,JHEP05(2019) 207 [1810.04619]

    2019 arXiv

  121. [132]

    Damour,Radiative contribution to classical gravitational scattering at the third order in G,Phys

    T. Damour,Radiative contribution to classical gravitational scattering at the third order in G,Phys. Rev. D102(2020) 124008 [2010.01641]

    2020 arXiv

  122. [133]

    Di Vecchia, C

    P. Di Vecchia, C. Heissenberg and R. Russo,Angular momentum of zero-frequency gravitons,JHEP08(2022) 172 [2203.11915]

    2022 arXiv

  123. [134]

    Guevara, A

    A. Guevara, A. Lupsasca, D. Skinner, A. Strominger and K. Weil,Single-minus gluon tree amplitudes are nonzero,2602.12176

  124. [135]

    Guevara, A

    A. Guevara, A. Lupsasca, D. Skinner, A. Strominger and K. Weil,Single-minus graviton tree amplitudes are nonzero,2603.04330

  125. [136]

    Francia and C

    D. Francia and C. Heissenberg,Two-Form Asymptotic Symmetries and Scalar Soft Theorems,Phys. Rev. D98(2018) 105003 [1810.05634]

    2018 arXiv

  126. [137]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett,Generalized Global Symmetries,JHEP 02(2015) 172 [1412.5148]

    2015 arXiv

  127. [138]

    Lake,Higher-form symmetries and spontaneous symmetry breaking,1802.07747

    E. Lake,Higher-form symmetries and spontaneous symmetry breaking,1802.07747

    arXiv

  128. [139]

    Tizzano,Comments on Symmetry Operators, Asymptotic Charges and Soft Theorems, 2604.06088

    L. Tizzano,Comments on Symmetry Operators, Asymptotic Charges and Soft Theorems, 2604.06088. – 57 –

    arXiv

  129. [140]

    Seiberg and S.-H

    N. Seiberg and S.-H. Shao,Exotic Symmetries, Duality, and Fractons in 2+1-Dimensional Quantum Field Theory,SciPost Phys.10(2021) 027 [2003.10466]. – 58 –

    2021 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.