BMS representations for generic supermomentum

Xavier Bekaert; Yannick Herfray

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BMS representations for generic supermomentum

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Pith's one-line read This paper proves that every BMS supermomentum splits uniquely into a hard momentum piece and a soft charge, yielding explicit wavefunctions for generic BMS particles with infinite Poincaré content and memory-carrying states.

desk verdict Serious paper with a repairable but load-bearing gap: Proposition 3.1 as stated fails for distributional soft charges, and the paper's own examples use them. read the letter →

arxiv 2505.05368 v1 pith:WUAQPVNZ submitted 2025-05-08 hep-th

Xavier Bekaert , Yannick Herfray This is my paper · ORCID

classification hep-th

keywords BMSgroupsupermomentumhard/softdecompositionunitaryirreduciblerepresentationscontinuous-spinmemoryeffectgravityvacuasoftcharges

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

The paper sets out to show that the unitary irreducible representations of the BMS group — the asymptotic symmetry group of flat spacetime — can be built explicitly and uniformly, one particle at a time, from a single canonical fact: every supermomentum $P(z,\bar z)$ splits uniquely and Lorentz-invariantly into a hard piece, a fixed nonlinear function of an ordinary four-momentum $p_\mu$, plus a soft charge $B_z^2B_{\bar z}^2N$ that measures how far the state departs from an ordinary Poincaré particle. From that split the authors write down explicit wavefunctions, inner products, and group actions for generic BMS particles, and show that restricting such a particle to any Poincaré subgroup always produces an infinite multiplet of ordinary particles — including, in the massless case, continuous-spin representations. They also display a normalizable BMS state whose average supermomentum carries a soft, memory-like piece that hard states cannot carry. If the construction is right, it matters because it gives representation theory a concrete route into gravitational infrared physics: supermomentum conservation forces soft charges into scattering, and two observers in different gravity vacua will disagree about which ordinary particles a given BMS state contains.

What carries the argument

The central object is the hard/soft decomposition of supermomenta, carried by the two-dimensional Paneitz operator $\eth^2\bar\eth^2 = B_z^2B_{\bar z}^2$ and the exact sequence $0\to R^{3,1}\hookrightarrow E[1]\xrightarrow{B_z^2B_{\bar z}^2} E[-3]\xrightarrow{\pi} (R^{3,1})^*\to 0$, which identifies soft supermomenta — those annihilating every translation — with the image of the Paneitz operator on the quotient $E[1]/R^{3,1}$ of supertranslations by translations. Proposition 3.1 converts this into a canonical pair $(p_\mu, N)$: the hard piece is fixed by the momentum, the soft charge $B_z^2B_{\bar z}^2N$ is fixed up to that momentum, and the whole decomposition is Lorentz-invariant but not linear. The non-linearity is the striking feature: adding two hard supermomenta produces a soft remainder built from the Weinberg soft factor $S=(q\cdot p_1)\ln|q\cdot p_1|+(q\cdot p_2)\ln|q\cdot p_2|-(q\cdot p_3)\ln|q\cdot p_3|$, which is why hard particles alone cannot conserve supermomentum. Feeding the decomposition into the Wigner-Mackey induction recipe then yields everything else: supermomentum orbits with little groups (stabilizers) given by the intersection of the Poincaré little group and the soft little group, explicit wavefunctions in the coordinates $(\omega,\zeta,\bar\zeta,\alpha,\beta,\bar\beta)$ of $SL(2,\mathbb C)$, and branching rules obtained by Fourier decomposing over the Euclidean little group, with the Fourier label $|\vec\pi|$ becoming the continuous-spin parameter.

What would settle it

Exhibit a supermomentum with two distinct Lorentz-invariant hard/soft decompositions (or a soft distribution outside the image of the Paneitz operator): Proposition 3.1 would fail and the classification built on it would collapse. Alternatively, find a unitary irreducible representation of BMS4 in the nuclear topology that is strictly ergodic — not induced from a supermomentum orbit — which would show the explicit wavefunctions miss some generic particles; a reader could also check whether the Hilbert-topology exhaustion proof extends, since the paper explicitly leaves that open. A lighter test targets the memory example: compute the full $L^2$ norm of the Section 9 state including subleading corrections in $\|B_z^2C\|_\infty/\omega$; if that norm diverges, the advertised memory state is not normalizable.

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Extended reading notes

Core claim

The paper's load-bearing claim is Proposition 3.1: every supermomentum $P(z,\bar z)\in E[-3]$ admits a unique, $SL(2,\mathbb C)$-invariant decomposition $P = \mathbb P + B_z^2B_{\bar z}^2 N$, where the hard part $\mathbb P$ is the specific nonlinear function of the momentum $p_\mu=\pi_\mu(P)$ written as $-(m^4/\pi)(p\cdot q(z,\bar z))^{-3}$ for massive momenta and $\omega\,\delta^{(2)}(z-\zeta,\bar z-\bar\zeta)$ for massless ones, and where the soft charge $B_z^2B_{\bar z}^2N$ is built from a shift of vacua $N\in E[1]/R^{3,1}$. Reading McCarthy's classification through this lens, the paper organizes BMS particles into hard (vanishing soft charge), soft (vanishing momentum), and generic (both nonzero) representations, and insists that generic ones — whose little group is typically trivial — are the physically relevant case. For those it provides explicit wavefunctions on $SL(2,\mathbb C)$, the action of supertranslations, and branching rules: a generic massless BMS particle becomes, in any Poincaré subgroup, one massless particle of every helicity plus a direct integral of continuous-spin representations, while a generic massive one becomes a tower of all spins with multiplicity $2j+1$. Finally the paper constructs a finite-norm BMS state built from a scalar field superposed with its own supertranslated copy in another gravity vacuum, and shows its average supermomentum acquires a soft part $B_{\bar z}^2\langle N\rangle$ — a memory effect that hard/Poincaré states cannot produce. The upshot, stated on the paper's own terms, is that BMS particles with nonzero soft charge are exactly as legitimate as ordinary hard particles, and that the gravitational infrared structure is encoded in the representation theory rather than added by hand.

Load-bearing premise

The construction assumes that every unitary irreducible representation of BMS4 is obtained by the standard Wigner-Mackey induction recipe — choose an $SL(2,\mathbb C)$-orbit of supermomenta, choose a unitary representation of the little group, build square-integrable sections — and the paper itself flags (Section 4.2, footnote 16) that this exhaustion is proven only in the Hilbert topology, not in the nuclear topology that is physically preferred; if additional, ergodic representations exist that escape the recipe, the explicit wavefunctions would not cover 'generic' BMS particles.

Editorial extensions

If this is right

Hard BMS representations coincide with Poincaré representations: their BMS and Poincaré little groups are equal, they do not branch under restriction to any Poincaré subgroup, and observers in different gravity vacua agree on their particle content.
Hard supermomenta do not add linearly, so hard particles alone can never conserve supermomentum in a scattering process; any BMS-invariant process must involve states with nonzero soft charge, and the soft charge produced by adding two hard particles is exactly the Weinberg soft factor.
A generic massless BMS particle restricts to any Poincaré subgroup as one massless particle of every helicity (all integer or all half-integer, multiplicity one) plus a direct integral of continuous-spin representations over all $\mu>0$; a generic massive one restricts to a tower of all spins $j$, each with multiplicity $2j+1$, all of the same mass.
The interpretation of a BMS state depends on the gravity vacuum: a state that is a single scalar in one vacuum is an infinite superposition of particles of all spins, including continuous spin, in another.
Generic BMS particles can encode gravitational memory: a specific normalizable state formed by superposing a scalar field with its supertranslated copy has average supermomentum with a soft, memory-like part that hard/Poincaré states cannot carry, because their $L^2$ norm forbids the required energy pole.

Reading between the lines

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

Reading the soft charge as the representation-theoretic counterpart of the usual dressed-state construction suggests a cleaner route to infrared-finite scattering: each BMS particle carries its own soft charge, and the Fock space built from such particles is automatically separable; the paper leaves the connection to explicit dressed amplitudes implicit.
The Lorentz-invariant distance between gravity vacua introduced in Corollary 8.3 is a natural control parameter for memory: one could test whether the memory amplitude of generic superpositions scales with that distance, an extension the paper only sketches at the end of Section 9.
Because every generic massless BMS particle branches to continuous-spin representations, any BMS-invariant S-matrix would evade the Coleman-Mandula theorem, whose finiteness assumptions the infinite Poincaré multiplet manifestly violates; whether a nontrivial S-matrix built on these multiplets actually exists is an open, testable question.
The Gaussian-regularized states of Section 6.5, which soften the $L^2$ norm enough to admit energy poles, offer a concrete mechanism for constructing infrared-finite amplitudes: checking whether their Fock-space amplitudes reproduce the standard soft theorems in the limit where the continuous-spin cutoff is removed would be a direct test of the whole picture.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know before you open it. First, the paper actually delivers on its abstract: explicit wavefunctions, branching rules, and a memory example for BMS UIRs. Second, the load-bearing decomposition (Prop. 3.1) is stated too broadly; as written, it fails for distributional supermomenta, and the authors' own example soft charges are distributional.

The genuinely new parts are the Lorentz-invariant hard/soft split, the completion of the massless branching analysis (Propositions 6.10, 6.11), and the normalizable BMS state with memory in Section 9. The appendices contain careful distributional identities and a real proof of the Paneitz exact sequence. That is reproducible math and should be credited.

Now the soft spot, and it is load-bearing. Proposition 3.1 claims every supermomentum decomposes uniquely as hard plus ∂²_z∂²_z̄ N with N a smooth conformal density of weight one modulo translations. But the massless hard piece is a δ, so for a supermomentum like δ + ∂²_z δ the soft part is ∂²_z δ. That annihilates translations, so it is soft, but it is not the Paneitz image of any smooth N; in spherical harmonics it has infinitely many nonvanishing high-ℓ modes, and the prospective N would have coefficients decaying like ℓ^{−3/2}, i.e., not smooth. The authors' own examples (6.20) and (6.21) are precisely of this type, and Appendix A.2 proves only a dual statement with N in a larger distributional dual space. So the proposition as stated, and the proof invoking Proposition 2.14, does not close the gap. The repair is likely straightforward—allow N to be a distribution modulo translations—but it changes the target space in Proposition 3.1 and will ripple through the orbit parametrization and wavefunction formulas. This is a major revision point, not a typo.

Other caveats are minor by comparison. The classification in the nuclear topology is still incomplete (footnote 16), and Sections 8 and 9 are informal, as the authors admit.

Who is this for? People working on asymptotic symmetries, infrared structure, or continuous-spin representations will get real value from the explicit branching rules and the memory construction, once the decomposition is fixed.

Recommendation: send it to peer review, but the referee should force a careful statement of Proposition 3.1 in the distributional setting before acceptance.

Referee Report

2 major / 4 minor

Summary. The paper revisits McCarthy's classification of unitary irreducible representations (UIRs) of the four-dimensional BMS group and aims to make the representations as explicit as possible. The main new ingredient is a Lorentz-invariant decomposition of supermomenta into a hard part, which is a nonlinear function of an ordinary four-momentum, and a soft part written as the Paneitz operator acting on a shift of vacuum. Using this decomposition, the authors give explicit wavefunctions, inner products, group actions, and branching rules for massless and massive BMS representations, including cases with non-trivial little groups. They argue that generic BMS particles branch to Poincaré subgroups into infinite towers of usual particles plus continuous-spin representations, that the wavefunctions reproduce the dependence on gravity vacua previously proposed in related work, and that a particular BMS state can carry an average supermomentum with a memory-like soft component. The paper is largely self-contained, with distributional identities and proofs collected in appendices, and it explicitly notes the open status of the full classification in the nuclear topology.

Significance. If the central decomposition and the explicit realizations are correct, the paper provides a valuable toolkit for BMS representation theory: it makes the hard/soft split of supermomenta concrete, supplies explicit formulas for wavefunctions and inner products in the massless and massive cases, and derives branching rules that are new or only implicit in McCarthy's work. The detailed appendices with Paneitz identities and distributional checks are a genuine strength, as is the careful treatment of boundary terms in the distributional identities for hard supermomenta. The connection between generic BMS particles, continuous-spin representations, gravity vacua, and the memory effect is physically suggestive and gives concrete calculable examples. However, the central decomposition as stated does not cover the distributional supermomenta that the paper itself uses in the examples, so the main technical claim needs repair before the rest of the construction can be regarded as established.

major comments (2)

[Section 3.1, Proposition 3.1; Section 2.2.2; Appendix A.2]
[Section 4.2, Theorem 4.3 and footnote 16]

minor comments (4)

[Section 3.1, equations (3.1)-(3.4)]
[Proposition 2.14]
[Section 6.4.2, Proposition 6.11]
[Footnote 16]

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the hard/soft decomposition and branching rules are derived from the Paneitz exact sequence and McCarthy's external classification, with self-citation [31] used only for contact/motivation in Section 8.

full rationale

This paper's derivation chain is self-contained with respect to its main claims. The hard/soft decomposition (Proposition 3.1) is not a definition of the soft charge N: it is obtained by projecting P to its momentum p via Proposition 2.10 and then invoking Proposition 2.14, which independently proves on the stated function spaces that every zero-momentum supermomentum is the Paneitz image of a supertranslation modulo translations, with proofs supplied in Appendices A.1-A.3. The classification of BMS UIRs (Theorem 4.3) and the branching rules (Theorem 4.7) are taken from McCarthy/Crampin as external results, and the paper explicitly flags the remaining nuclear-topology gap in the exhaustive proof; this is an acknowledged assumption, not a circular one. The wavefunction constructions and branching computations in Sections 6-7 are concrete consequences of those external inputs rather than restatements of them. The principal self-citation is [31] (Bekaert-Donnay-Herfray), used in Section 8 for Fourier-transform pictures and in the introduction for motivation; it is not used to prove the decomposition, uniqueness, or branching rules, and the branching content is independently derived in Sections 6-7. The skeptical concern that Proposition 3.1 may fail for distributional soft charges such as ∂_z^2 δ^(2) is a mathematical correctness issue about the image of the Paneitz operator in the distributional category, not a circularity: it challenges whether the stated theorem is true, not whether the theorem was assumed as its own conclusion. No fitted parameter is renamed as a prediction, and no load-bearing premise is justified only by an author self-citation. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

No empirical fit parameters are introduced; masses, helicities, and soft charges are representation labels. The derivations rely on prior classification theorems and functional-analytic conventions, listed as axioms. The one invented entity, the soft charge, is a definitional mathematical object rather than a postulated physical particle.

assumptions (3)

domain assumption All unitary irreducible representations of BMS4 are induced representations from supermomentum orbits (Mackey-McCarthy classification).
Theorem 4.3 states this classification. The paper flags that completeness is proven only in the Hilbert topology, not the physically preferred nuclear topology (Section 4.2 and footnote 16). The paper's identification of generic BMS particles relies on this theorem.
standard math Projective unitary representations of BMS4 lift to linear unitary representations of the universal covering group.
Invoked in Section 2.2.1 and in the proof discussion of Theorem 4.3, following McCarthy [16]. It allows treating projective representations as ordinary ones without repeating the lifting argument.
domain assumption The space of gravity vacua is the homogeneous space BMS4/ISO0(3,1).
Definition 8.2 imports this from Ashtekar [5]. It is used to interpret branching to a Poincare subgroup as reading off field content in a chosen gravity vacuum.

invented entities (1)

soft charge N(z,zbar)
purpose: Labels the soft part of a supermomentum under the hard/soft decomposition; distinguishes hard, soft, and generic BMS representations.
It is defined purely by Proposition 3.1 within the paper's framework. Although it is tied to infrared and memory observables in Section 9, the paper provides no externally validated measurement of this quantity independent of its own construction.

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Pith. "Pith review of BMS representations for generic supermomentum." pith.science (2026). https://pith.science/paper/WUAQPVNZ

@misc{pith2026250505368,
  author       = {Pith},
  title        = {Pith review of: BMS representations for generic supermomentum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WUAQPVNZ}},
  note         = {Machine review of arXiv:2505.05368}
}

read the original abstract

We revisit the classification, and give explicit realisations, of unitary irreducible representations of the BMS group. As compared to McCarthy's seminal work, we make use of a unique, Lorentz-invariant, decomposition of supermomenta into a hard and a soft piece, that we introduce and properly define, to investigate the extent to which generic representations depart from usual Poincar\'e particles and highlight their relations to gravitational infrared physics. We insist on making wavefunctions as explicit as possible. Similarly, we explain how branching to a Poincar\'e subgroup works in practice: this is physically relevant because this amounts to reading off the field content of a given BMS state in terms of a choice of gravity vacuum. In particular, we emphasise how different gravity vacua differ in their interpretation of the same BMS state, here again providing concrete examples as well as the general procedure. Finally, we demonstrate on an example that generic BMS particles are flexible enough to encode memory, as opposed to usual Poincar\'e particles.

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Reference graph

Works this paper leans on

103 extracted references · 30 canonical work pages · cited by 4 Pith papers

[1]

Gravitational waves in general relativity, VII. Waves from axi-symmetric isolated system,

H. Bondi, M. G. J. Van der Burg, and A. W. K. Metzner, “Gravitational waves in general relativity, VII. Waves from axi-symmetric isolated system,” Proc. R. Soc. (London) A 269 (1962) no. 1336, 21–52

1962
[2]

Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times,

R. K. Sachs, “Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times,” Proc. Roy. Soc. Lond. A 270 (1962) 103–126

1962
[3]

Asymptotic structure of space-time,

R. Geroch, “Asymptotic structure of space-time,” in Asymptotic Structure of Space-Time , F. P. Esposito and L. Witten, eds., pp. 1–105. Springer US, Boston, MA, 1977

1977
[4]

Conformal Carroll groups and BMS symmetry,

C. Duval, G. W. Gibbons, and P. A. Horvathy, “Conformal Carroll groups and BMS symmetry,” Class. Quant. Grav. 31 (2014) 092001, arXiv:1402.5894 [gr-qc]

arXiv 2014
[5]

Ashtekar, Asymptotic Quantization: Based on 1984 Naples Lectures,

A. Ashtekar, Asymptotic Quantization: Based on 1984 Naples Lectures, . Bibliopolis, 1987

1984
[6]

On BMS Invariance of Gravitational Scattering,

A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 07 (2014) 152, arXiv:1312.2229 [hep-th]

arXiv 2014
[7]

BMS supertranslations and Weinberg’s soft graviton theorem,

T. He, V. Lysov, P. Mitra, and A. Strominger, “BMS supertranslations and Weinberg’s soft graviton theorem,” JHEP 05 (2015) 151, arXiv:1401.7026 [hep-th]

arXiv 2015
[8]

Representations of the bondi-metzner-sachs group

P. J. McCarthy, “Representations of the bondi-metzner-sachs group.” preprint of International Center of Theoretical Physics (Trieste, Italy) no IC/71/144, nov, 1971

1971

Show all 103 references

[9]

Asymptotically flat space-times and elementary particles,

P. J. M. McCarthy, “Asymptotically flat space-times and elementary particles,” Phys. Rev. Lett. 29 (1972) 817–819

1972
[10]

Representations of the bondi-metzner-sachs group. i. determination of the representations,

P. J. McCarthy, “Representations of the bondi-metzner-sachs group. i. determination of the representations,” Proc. R. Soc. (London) A 330 (1972) no. 1583, 517–535

1972
[11]

Representations of the bondi-metzner-sachs group. ii. properties and classification of the representations,

P. J. McCarthy, “Representations of the bondi-metzner-sachs group. ii. properties and classification of the representations,” Proc. R. Soc. (London) A 333 (1973) no. 1594, 317–336

1973
[12]

Representations of the bondi-metzner-sachs group. iii. poincare spin multiplicities and irreducibility,

P. J. McCarthy and M. Crampin, “Representations of the bondi-metzner-sachs group. iii. poincare spin multiplicities and irreducibility,” Proc. R. Soc. (London) A 335 (1973) no. 1602, 301–311

1973
[13]

Physical significance of the topology of the bondi-metzner-sachs group,

M. Crampin and P. J. McCarthy, “Physical significance of the topology of the bondi-metzner-sachs group,” Phys. Rev. Lett. 33 (1974) 547–550

1974
[14]

The bondi-metzner-sachs group in the nuclear topology,

P. J. McCarthy, “The bondi-metzner-sachs group in the nuclear topology,” Proc. R. Soc. (London) A 343 (1975) no. 1635, 489–523

1975
[15]

Representations of the bondi-metzner-sachs group. iv. cantoni representations are induced,

M. Crampin and P. J. McCarthy, “Representations of the bondi-metzner-sachs group. iv. cantoni representations are induced,” Proc. R. Soc. (London) A 351 (1976) no. 1664, 55–70

1976
[16]

Lifting of projective representations of the bondi—metzner—sachs group,

P. J. McCarthy, “Lifting of projective representations of the bondi—metzner—sachs group,” Proc. R. Soc. (London) A 358 (1978) 141–171

1978
[17]

Hyperfunctions and asymptotic symmetries,

P. J. McCarthy, “Hyperfunctions and asymptotic symmetries,” Proc. R. Soc. (London) A 358 (1978) no. 1695, 495–498

1978
[18]

A class of representations of the generalized bondi—metzner group,

V. Cantoni, “A class of representations of the generalized bondi—metzner group,” J. Math. Phys. 7 (1966) 1361–1364

1966
[19]

Induction of representations of the generalized Bondi-Metzner group,

V. Cantoni, “Induction of representations of the generalized Bondi-Metzner group,” Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 43 (1967) 30–37

1967
[20]

Reduction of some representations of the generalized bondi-metzner group,

V. Cantoni, “Reduction of some representations of the generalized bondi-metzner group,” J. Math. Phys. 8 (1967) 1700–1706

1967
[21]

Continuous spins in the bondi-metzner-sachs group of asymptotic symmetry in general relativity,

L. Girardello and G. Parravicini, “Continuous spins in the bondi-metzner-sachs group of asymptotic symmetry in general relativity,” Phys. Rev. Lett. 32 (1974) 565–568. – 68 –

1974
[22]

Approaching quantum gravity via the representation theory of the BMS group and its variants. Some conceptual issues and a conjecture,

E. Melas, “Approaching quantum gravity via the representation theory of the BMS group and its variants. Some conceptual issues and a conjecture,” Nucl. Phys. B Proc. Suppl. 104 (2002) 212–216

2002
[23]

Open problems and results in the group theoretic approach to quantum gravity via the bms group and its generalizations,

E. Melas, “Open problems and results in the group theoretic approach to quantum gravity via the bms group and its generalizations,” J. Phys. Conf. Ser. 283 (2011) no. 1, 012023

2011
[24]

First results on the representation theory of the Ultrahyperbolic BMS group UHB(2, 2),

E. Melas, “First results on the representation theory of the Ultrahyperbolic BMS group UHB(2, 2),” J. Phys. Conf. Ser. 670 (2016) no. 1, 012035

2016
[25]

Notes on the BMS group in three dimensions: I. Induced representations,

G. Barnich and B. Oblak, “Notes on the BMS group in three dimensions: I. Induced representations,” JHEP 06 (2014) 129, arXiv:1403.5803 [hep-th]

2014 arXiv
[26]

Notes on the BMS group in three dimensions: II. Coadjoint representation,

G. Barnich and B. Oblak, “Notes on the BMS group in three dimensions: II. Coadjoint representation,” JHEP 03 (2015) 033, arXiv:1502.00010 [hep-th]

2015 arXiv
[27]

Oblak, BMS Particles in Three Dimensions

B. Oblak, BMS Particles in Three Dimensions . PhD thesis, U. Brussels, Brussels U., 2016. arXiv:1610.08526 [hep-th]

2016 arXiv
[28]

On the representation theory of the Bondi–Metzner–Sachs group and its variants in three space–time dimensions,

E. Melas, “On the representation theory of the Bondi–Metzner–Sachs group and its variants in three space–time dimensions,” J. Math. Phys. 58 (2017) no. 7, 071705, arXiv:1703.05980 [math-ph]

2017 arXiv
[29]

On the definition of the spin charge in asymptotically-flat spacetimes,

L. Freidel, S. F. Moosavian, and D. Pranzetti, “On the definition of the spin charge in asymptotically-flat spacetimes,” arXiv:2403.19547 [hep-th]

arXiv
[30]

A Canonical realization of the BMS algebra,

G. Longhi and M. Materassi, “A Canonical realization of the BMS algebra,” J. Math. Phys. 40 (1999) 480–500, arXiv:hep-th/9803128

1999 arXiv
[31]

BMS particles,

X. Bekaert, L. Donnay, and Y. Herfray, “BMS particles,” arXiv:2412.06002 [hep-th]
[32]

Coadjoint representation of the BMS group on celestial Riemann surfaces,

G. Barnich and R. Ruzziconi, “Coadjoint representation of the BMS group on celestial Riemann surfaces,” JHEP 06 (2021) 079, arXiv:2103.11253 [gr-qc]

2021 arXiv
[33]

BMS flux algebra in celestial holography,

L. Donnay and R. Ruzziconi, “BMS flux algebra in celestial holography,” JHEP 11 (2021) 040, arXiv:2108.11969 [hep-th]

2021 arXiv
[34]

Asymptotic Symmetries in Gravitational Theory,

R. K. Sachs, “Asymptotic Symmetries in Gravitational Theory,” Phys. Rev. 128 (1962) no. 6, 2851–2864

1962
[35]

Asymptotic symmetries of gravity and soft theorems for massive particles,

M. Campiglia and A. Laddha, “Asymptotic symmetries of gravity and soft theorems for massive particles,” JHEP 12 (2015) 094, arXiv:1509.01406 [hep-th]

2015 arXiv
[36]

BMS symmetry, soft particles and memory,

A. Chatterjee and D. A. Lowe, “BMS symmetry, soft particles and memory,” Class. Quant. Grav. 35 (2018) no. 9, 094001, arXiv:1712.03211 [hep-th]

2018 arXiv
[37]

Infrared Divergences in QED, Revisited,

D. Kapec, M. Perry, A.-M. Raclariu, and A. Strominger, “Infrared Divergences in QED, Revisited,” Phys. Rev. D 96 (2017) no. 8, 085002, arXiv:1705.04311 [hep-th]

2017 arXiv
[38]

BMS Supertranslation Symmetry Implies Faddeev-Kulish Amplitudes,

S. Choi and R. Akhoury, “BMS Supertranslation Symmetry Implies Faddeev-Kulish Amplitudes,” JHEP 02 (2018) 171, arXiv:1712.04551 [hep-th]

2018 arXiv
[39]

Asymptotic conditions and infrared divergences in quantum electrodynamics,

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

1970
[40]

Gravitational Memory, BMS Supertranslations and Soft Theorems,

A. Strominger and A. Zhiboedov, “Gravitational Memory, BMS Supertranslations and Soft Theorems,” JHEP 01 (2016) 086, arXiv:1411.5745 [hep-th]

2016 arXiv
[41]

Note on the Bondi-Metzner-Sachs group,

E. T. Newman and R. Penrose, “Note on the Bondi-Metzner-Sachs group,” J. Math. Phys. 7 (1966) 863–870

1966
[42]

Edth - a differential operator on the sphere,

M. Eastwood and P. Tod, “Edth - a differential operator on the sphere,” Math. Proc. of Cambridge Phil. Soc. 92 (1982) no. 2, 317–330. – 69 –

1982
[43]

Celestial w1`8 Symmetries from Twistor Space,

T. Adamo, L. Mason, and A. Sharma, “Celestial w1`8 Symmetries from Twistor Space,” SIGMA 18 (2022) 016, arXiv:2110.06066 [hep-th]

2022 arXiv
[44]

Graviton scattering in self-dual radiative space-times,

T. Adamo, L. Mason, and A. Sharma, “Graviton scattering in self-dual radiative space-times,” Class. Quant. Grav. 40 (2023) no. 9, 095002, arXiv:2203.02238 [hep-th]

2023 arXiv
[45]

I. M. Gelfand, M. I. Graev, and N. Y. Vilenkin, Generalized Functions Vol 5 Integral Geometry And Representation Theory. AMS Chelsea Publishing. Academic Press, New York, NY, 1966

1966
[46]

A quartic conformally covariant differential operator for arbitrary pseudo-riemannian manifolds (summary),

S. Paneitz, “A quartic conformally covariant differential operator for arbitrary pseudo-riemannian manifolds (summary),” SIGMA 4 (2008) 036, arXiv:0803.4331 [math.DG]

2008 arXiv
[47]

Asymptotic freedom in extended conformal supergravities,

E. S. Fradkin and A. A. Tseytlin, “Asymptotic freedom in extended conformal supergravities,” Phys. Lett. B 110 (1982) 117–122. [Erratum: Phys.Lett. B 126 (1983)]

1982
[48]

I. M. Gelfand and G. E. Shilov, Generalized Functions Vol 1 Properties And Operations . AMS Chelsea Publishing. Academic Press, New York, NY, 1964. Trans. from the Russian, Moscow, 1958

1964
[49]

Green’s functions of the edh operators,

J. Ivancovich, C. Kozameh, and E. T. Newman, “Green’s functions of the edh operators,” Journal of mathematical physics 30 (1989) no. 1, 45–52

1989
[50]

An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity,

S. N. Curry and A. R. Gover, “An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity,” in Asymptotic Analysis in General Relativity , T. Daud´ e, D. H¨ afner, and J.-P. Nicolas, eds., London Mathematical Society Lecture No...

2018
[51]

On Unitary Representations of the Inhomogeneous Lorentz Group,

E. P. Wigner, “On Unitary Representations of the Inhomogeneous Lorentz Group,” Annals Math. 40 (1939) 149–204

1939
[52]

G. W. Mackey, Induced representations of groups and quantum mechanics . W. A. Benjamin, New York, USA, 1968

1968
[53]

Unitary representations of semi-direct product groups with infinite dimensional abelian normal subgroup,

A. Piard, “Unitary representations of semi-direct product groups with infinite dimensional abelian normal subgroup,” Rep. Math. Phys. 11 (1977) no. 2, 259–278

1977
[54]

Representations of the bondi-metzner-sachs group with the hilbert topology,

A. Piard, “Representations of the bondi-metzner-sachs group with the hilbert topology,” Rep. Math. Phys. 11 (1977) no. 2, 279–283

1977
[55]

Borel multipliers for the bondi–metzner–sachs group,

U. Cattaneo, “Borel multipliers for the bondi–metzner–sachs group,” J. Math. Phys. 20 (1979) no. 11, 2257–2263

1979
[56]

On the projective representations of the bondi-metzner-sachs group,

Y. K. Lau and X. N. Wu, “On the projective representations of the bondi-metzner-sachs group,” Proceedings: Mathematical, Physical and Engineering Sciences 457 (2001) no. 2006, 453–469

2001
[57]

Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations . Springer New York, 2010

2010
[58]

Quantized gravitational theory and internal symmetries,

A. Komar, “Quantized gravitational theory and internal symmetries,” Phys. Rev. Lett. 15 (1965) 76–78

1965
[59]

All Possible Symmetries of the S Matrix,

S. R. Coleman and J. Mandula, “All Possible Symmetries of the S Matrix,” Phys. Rev. 159 (1967) 1251–1256

1967
[60]

Weinberg, The quantum theory of fields

S. Weinberg, The quantum theory of fields. Vol. 3: Supersymmetry . Cambridge University Press, 2013

2013
[61]

Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory

A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory . Princeton University Press, 2017. arXiv:1703.05448 [hep-th]

2017 arXiv
[62]

Bridging Carrollian and celestial holography,

L. Donnay, A. Fiorucci, Y. Herfray, and R. Ruzziconi, “Bridging Carrollian and celestial holography,” Phys. Rev. D 107 (2023) no. 12, 126027, arXiv:2212.12553 [hep-th]

2023 arXiv
[63]

Massless scalars and higher-spin BMS in any dimension,

X. Bekaert and B. Oblak, “Massless scalars and higher-spin BMS in any dimension,” JHEP 11 – 70 – (2022) 022, arXiv:2209.02253 [hep-th]

2022 arXiv
[64]

Carrollian Conformal Fields and Flat Holography,

K. Nguyen and P. West, “Carrollian Conformal Fields and Flat Holography,” Universe 9 (2023) no. 9, 385, arXiv:2305.02884 [hep-th]

2023 arXiv
[65]

Null to time-like infinity Green’s functions for asymptotic symmetries in Minkowski spacetime,

M. Campiglia, “Null to time-like infinity Green’s functions for asymptotic symmetries in Minkowski spacetime,” JHEP 11 (2015) 160, arXiv:1509.01408 [hep-th]

2015 arXiv
[66]

Massive carrollian fields at timelike infinity,

E. Have, K. Nguyen, S. Prohazka, and J. Salzer, “Massive carrollian fields at timelike infinity,” JHEP 07 (2024) 054, arXiv:2402.05190 [hep-th]

2024 arXiv
[67]

Ti and Spi, Carrollian extended boundaries at timelike and spatial infinity,

J. Borthwick, M. Chantreau, and Y. Herfray, “Ti and Spi, Carrollian extended boundaries at timelike and spatial infinity,” arXiv:2412.15996 [gr-qc]

arXiv
[68]

Einstein’s equations near spatial infinity,

R. Beig and B. G. Schmidt, “Einstein’s equations near spatial infinity,” Commun. Math. Phys. 87 (1982) no. 1, 65–80

1982
[69]

Carrollian and celestial spaces at infinity,

J. Figueroa-O’Farrill, E. Have, S. Prohazka, and J. Salzer, “Carrollian and celestial spaces at infinity,” JHEP 09 (2022) 007, arXiv:2112.03319 [hep-th]

2022 arXiv
[70]

Asymptotic symmetries of projectively compact order one Einstein manifolds,

J. Borthwick and Y. Herfray, “Asymptotic symmetries of projectively compact order one Einstein manifolds,” Phil. Trans. Roy. Soc. Lond. A 382 (2024) no. 2267, 20230042, arXiv:2309.09962 [math.DG]

2024 arXiv
[71]

Projective and Carrollian geometry at time/space-like infinity on projectively compact Ricci flat Einstein manifolds,

J. Borthwick and Y. Herfray, “Projective and Carrollian geometry at time/space-like infinity on projectively compact Ricci flat Einstein manifolds,” Geom. Dedicata 219 (2025) 12, arXiv:2406.01800 [math.DG]

2025 arXiv
[72]

A unified treatment of null and spatial infinity in general relativity. I - Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity,

A. Ashtekar and R. O. Hansen, “A unified treatment of null and spatial infinity in general relativity. I - Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity,” J. Math. Phys. 19 (1978) 1542–1566

1978
[73]

Spatial infinity as a boundary of space-time,

A. Ashtekar and J. D. Romano, “Spatial infinity as a boundary of space-time,” Class. Quant. Grav. 9 (1992) 1069–1100

1992
[74]

The BMS4 algebra at spatial infinity,

C. Troessaert, “The BMS4 algebra at spatial infinity,” Class. Quant. Grav. 35 (2018) no. 7, 074003, arXiv:1704.06223 [hep-th]

2018 arXiv
[75]

Hamiltonian structure and asymptotic symmetries of the Einstein-Maxwell system at spatial infinity,

M. Henneaux and C. Troessaert, “Hamiltonian structure and asymptotic symmetries of the Einstein-Maxwell system at spatial infinity,” JHEP 07 (2018) 171, arXiv:1805.11288 [gr-qc]

2018 arXiv
[76]

BMS Group at Spatial Infinity: the Hamiltonian (ADM) approach,

M. Henneaux and C. Troessaert, “BMS Group at Spatial Infinity: the Hamiltonian (ADM) approach,” JHEP 03 (2018) 147, arXiv:1801.03718 [gr-qc]

2018 arXiv
[77]

The asymptotic structure of gravity at spatial infinity in four spacetime dimensions,

M. Henneaux and C. Troessaert, “The asymptotic structure of gravity at spatial infinity in four spacetime dimensions,” Proc. Steklov Inst. Math. 309 (2020) 127, arXiv:1904.04495 [hep-th]

2020 arXiv
[78]

Supertranslations at timelike infinity,

S. Chakraborty, D. Ghosh, S. J. Hoque, A. Khairnar, and A. Virmani, “Supertranslations at timelike infinity,” JHEP 02 (2022) 022, arXiv:2111.08907 [hep-th]

2022 arXiv
[79]

An asymptotic framework for gravitational scattering,

G. Comp` ere, S. E. Gralla, and H. Wei, “An asymptotic framework for gravitational scattering,” Class. Quant. Grav. 40 (2023) no. 20, 205018, arXiv:2303.17124 [gr-qc]

2023 arXiv
[80]

Weinberg, The Quantum theory of fields

S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations . Cambridge University Press, 6, 2005

2005
[81]

The unitary representations of the Poincar \’e group in any spacetime dimension,

X. Bekaert and N. Boulanger, “The unitary representations of the Poincar \’e group in any spacetime dimension,” SciPost Phys. Lect. Notes 30 (2021) 1, arXiv:hep-th/0611263

2021 arXiv
[82]

Elementary particles with continuous spin,

X. Bekaert and E. D. Skvortsov, “Elementary particles with continuous spin,” Int. J. Mod. Phys. A 32 (2017) no. 23n24, 1730019, arXiv:1708.01030 [hep-th]

2017 arXiv
[83]

Infrared finite scattering theory in quantum field theory and quantum gravity,

K. Prabhu, G. Satishchandran, and R. M. Wald, “Infrared finite scattering theory in quantum field theory and quantum gravity,” Phys. Rev. D 106 (2022) no. 6, 066005, arXiv:2203.14334 – 71 – [hep-th]

2022 arXiv
[84]

Infrared finite scattering theory: scattering states and representations of the BMS group,

K. Prabhu and G. Satishchandran, “Infrared finite scattering theory: scattering states and representations of the BMS group,” JHEP 08 (2024) 055, arXiv:2402.00102 [hep-th]

2024 arXiv
[85]

Interactions of Particles with

P. Schuster, N. Toro, and K. Zhou, “Interactions of Particles with ”Continuous Spin” Fields,” JHEP 04 (2023) 010, arXiv:2303.04816 [hep-th]

2023 arXiv
[86]

Tensor harmonics on the 3-sphere,

R. T. Jantzen, “Tensor harmonics on the 3-sphere,” J. Math. Phys. 19 (1978) no. 5, 1163–1172

1978
[87]

The peter–weyl theorem & harmonic analysis on sn,

L. Nashabeh, “The peter–weyl theorem & harmonic analysis on sn,” Columbia J. of Undergrad. Math. 1 (2024) no. 1, 70–98

2024
[88]

Group theory and the hydrogen atom (i),

M. Bander and C. Itzykson, “Group theory and the hydrogen atom (i),” Rev. Mod. Phys. 38 (1966) 330–345

1966
[89]

Shiraishi, Probing the Early Universe with the CMB Scalar, Vector and Tensor Bispectrum

M. Shiraishi, Probing the Early Universe with the CMB Scalar, Vector and Tensor Bispectrum . Springer Theses. Springer, 2013. arXiv:1210.2518 [astro-ph.CO]

2013 arXiv
[90]

A. O. Barut and R. Raczka, Theory of group representations and applications . Polish Scientific Publishers, Warsaw, 1977

1977
[91]

Geometry and Physics of Null Infinity,

A. Ashtekar, “Geometry and Physics of Null Infinity,” arXiv:1409.1800 [gr-qc]

arXiv
[92]

Asymptotic shear and the intrinsic conformal geometry of null-infinity,

Y. Herfray, “Asymptotic shear and the intrinsic conformal geometry of null-infinity,” J. Math. Phys. 61 (2020) no. 7, 072502, arXiv:2001.01281 [gr-qc]

2020 arXiv
[93]

Tractor Geometry of Asymptotically Flat Spacetimes,

Y. Herfray, “Tractor Geometry of Asymptotically Flat Spacetimes,” Annales Henri Poincare 23 (2022) no. 9, 3265–3310, arXiv:2103.10405 [gr-qc]

2022 arXiv
[94]

Carrollian manifolds and null infinity: a view from Cartan geometry,

Y. Herfray, “Carrollian manifolds and null infinity: a view from Cartan geometry,” Class. Quant. Grav. 39 (2022) no. 21, 215005, arXiv:2112.09048 [gr-qc]

2022 arXiv
[95]

Shadows and soft exchange in celestial CFT,

D. Kapec and P. Mitra, “Shadows and soft exchange in celestial CFT,” Phys. Rev. D 105 (2022) no. 2, 026009, arXiv:2109.00073 [hep-th]

2022 arXiv
[96]

Soft scalars and the geometry of the space of celestial conformal field theories,

D. Kapec, Y. T. A. Law, and S. A. Narayanan, “Soft scalars and the geometry of the space of celestial conformal field theories,” Phys. Rev. D 107 (2023) no. 4, 046024, arXiv:2205.10935 [hep-th]

2023 arXiv
[97]

Soft particles and infinite-dimensional geometry,

D. Kapec, “Soft particles and infinite-dimensional geometry,” Class. Quant. Grav. 41 (2024) no. 1, 015001, arXiv:2210.00606 [hep-th]

2024 arXiv
[98]

Infrared finite scattering theory: Amplitudes and soft theorems,

K. Prabhu and G. Satishchandran, “Infrared finite scattering theory: Amplitudes and soft theorems,” Phys. Rev. D 110 (2024) no. 8, 085022, arXiv:2402.18637 [hep-th]

2024 arXiv
[99]

Null infinity, the BMS group and infrared issues,

A. Ashtekar, M. Campiglia, and A. Laddha, “Null infinity, the BMS group and infrared issues,” Gen. Rel. Grav. 50 (2018) no. 11, 140–163, arXiv:1808.07093 [gr-qc]

2018 arXiv
[100]

Hungerford, Algebra

T. Hungerford, Algebra. Springer, eighth ed., 2003

2003
[101]

Tung, Group Theory in Physics

W.-K. Tung, Group Theory in Physics . World Scientific, 1985

1985
[102]

B. P. Rynne and M. A. Youngson, Linear Functional Analysis. Springer London, 2008

2008
[103]

P. R. Halmos, A Hilbert Space Problem Book . Springer New York, 1982. – 72 –

1982

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Gravitational waves in general relativity, VII. Waves from axi-symmetric isolated system,

H. Bondi, M. G. J. Van der Burg, and A. W. K. Metzner, “Gravitational waves in general relativity, VII. Waves from axi-symmetric isolated system,” Proc. R. Soc. (London) A 269 (1962) no. 1336, 21–52

1962

[2] [2]

Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times,

R. K. Sachs, “Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times,” Proc. Roy. Soc. Lond. A 270 (1962) 103–126

1962

[3] [3]

Asymptotic structure of space-time,

R. Geroch, “Asymptotic structure of space-time,” in Asymptotic Structure of Space-Time , F. P. Esposito and L. Witten, eds., pp. 1–105. Springer US, Boston, MA, 1977

1977

[4] [4]

Conformal Carroll groups and BMS symmetry,

C. Duval, G. W. Gibbons, and P. A. Horvathy, “Conformal Carroll groups and BMS symmetry,” Class. Quant. Grav. 31 (2014) 092001, arXiv:1402.5894 [gr-qc]

arXiv 2014

[5] [5]

Ashtekar, Asymptotic Quantization: Based on 1984 Naples Lectures,

A. Ashtekar, Asymptotic Quantization: Based on 1984 Naples Lectures, . Bibliopolis, 1987

1984

[6] [6]

On BMS Invariance of Gravitational Scattering,

A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 07 (2014) 152, arXiv:1312.2229 [hep-th]

arXiv 2014

[7] [7]

BMS supertranslations and Weinberg’s soft graviton theorem,

T. He, V. Lysov, P. Mitra, and A. Strominger, “BMS supertranslations and Weinberg’s soft graviton theorem,” JHEP 05 (2015) 151, arXiv:1401.7026 [hep-th]

arXiv 2015

[8] [8]

Representations of the bondi-metzner-sachs group

P. J. McCarthy, “Representations of the bondi-metzner-sachs group.” preprint of International Center of Theoretical Physics (Trieste, Italy) no IC/71/144, nov, 1971

1971