Irreducible Representations of BMS Group

• Irreducible representations of the BMS group are infinite-dimensional induced structures characterized by supertranslation and superrotation quantum numbers that extend the Poincaré framework.
• The construction employs Mackey’s induction method to classify supermomentum orbits, with little groups like SU(2) and E(2) distinguishing massive from massless cases.
• These representations underpin our understanding of gravitational scattering, infrared physics, and memory effects by linking hard momentum with soft gravitational modes.

The Bondi–Metzner–Sachs (BMS) group and its infinite-dimensional symmetry algebra provide the universal asymptotic symmetry structure for asymptotically flat spacetimes. The unitary irreducible representations (UIRs) of the BMS group—called “BMS particles”—generalize the familiar notion of Poincaré particles, incorporating an infinite tower of “supermomentum” quantum numbers associated with supertranslation and, in extended generalizations, superrotation symmetries. The study and classification of these irreducible representations relies on Mackey–Wigner induction from little group orbits in the infinite-dimensional dual of the supertranslation algebra, and is crucial for a complete description of gravitational scattering, infrared physics, and the asymptotic phase space in both three and four spacetime dimensions.

1. Algebraic Structure of the BMS Group

The four-dimensional BMS group is realized as the semidirect product

$$\mathrm{BMS}_4 = \mathrm{SL}(2,\mathbb{C}) \ltimes C^\infty(S^2)$$

where $\mathrm{SL}(2,\mathbb{C})$ is the Lorentz group acting on the celestial sphere $S^2$ via Möbius transformations, and $C^\infty(S^2)$ is the abelian group of smooth “supertranslations.” The BMS Lie algebra is generated by Lorentz generators $J_{ab}$ ($[J_{ab}, J_{cd}]$ closes as usual) and an infinite tower of supertranslations $T[f]$, one for each function $f(\theta, \phi)$ on $S^2$, with

$$[T[f], T[g]] = 0, \quad [J_{ab}, T[f]] = i\, T[\mathcal{L}_{ab}f]$$

where $\mathcal{L}_{ab}$ is the standard Lorentz action on functions on the sphere (Chatterjee et al., 2017).

In three dimensions, the BMS group is analogously constructed as

$$\mathrm{BMS}_3 = \mathrm{Diff}^+(S^1) \ltimes \mathrm{Vect}(S^1)$$

where $\mathrm{Diff}^+(S^1)$ denotes orientation-preserving diffeomorphisms of the circle (superrotations) and $\mathrm{Vect}(S^1)$ their abelian vector-field ideal (supertranslations) (Amith et al., 20 Feb 2025).

2. Supermomentum Orbits and Little Groups

BMS unitary irreducible representations are constructed by induction from orbits in the dual supertranslation space. For $\mathrm{BMS}_4$, the dual of the supertranslation algebra consists of densities $P(z,\bar z)$ with transformation properties under $\mathrm{SL}(2, \mathbb{C})$, which generalize the notion of classical four-momentum:

$$\pi_\mu(P) = \int_{S^2} d^2z\, q_\mu(z, \bar z)\, P(z, \bar z)$$

where $q_\mu(z, \bar z)$ is a set of basis functions embedding translations into the supertranslation algebra. Orbits of the Lorentz group in the space of $P(z, \bar z)$ are labeled by the values of $\pi_\mu$ (the “hard” part), and the infinite set of additional “supermomentum” quantum numbers (the “soft” part) (Bekaert et al., 8 May 2025, Bekaert et al., 2024).

The little group $\ell_P \subseteq \mathrm{SL}(2, \mathbb{C})$ is the stabilizer of a given $P(z, \bar z)$. Its structure depends on the orbit:

• Massive: For $p^2 > 0$, $\ell_P \cong \mathrm{SU}(2)$.
• Massless: For $p^2 = 0$, “hard” orbits have $\ell_P \cong E(2)$ (the 2D Euclidean group).
• Soft/zero-momentum: For $\pi_\mu = 0$ but $P \ne 0$, possible little groups are $\mathrm{SL}(2, \mathbb{R})$ or $U(1)$, depending on the detailed support of $P(z, \bar z)$ (Chatterjee et al., 2017, Bekaert et al., 2024).

In three dimensions, the supermomentum is a function $p(\varphi)$ on the circle, and the classification of coadjoint orbits reduces to the structure of orbits in the Virasoro dual. The little groups are $U(1)$ (massive case), higher covers of $SL(2,\mathbb{R})$ (exceptional/vacuum-like), or trivial/finitely generated subgroups (massless/tachyonic) (Barnich et al., 2015, Amith et al., 20 Feb 2025, Melas, 2021).

3. Induced Representation Construction

The general Mackey–McCarthy construction proceeds as follows:

1. Choose an orbit: $O_P = \mathrm{SL}(2, \mathbb{C})/\ell_P$ in $E[-3]$, labeled by $\pi_\mu$ and the soft supermomentum data.
2. Pick a little group representation: Select an irrep $\rho: \ell_P \to U(V)$.
3. Build carrier space: Carrier space consists of $\rho$-equivariant functions on $O_P$,

$$\mathcal{H} = \{ \Phi: \mathrm{SL}(2,\mathbb{C}) \to V \mid \Phi(gb) = \rho(b^{-1}) \Phi(g),\ b \in \ell_P \}$$

with invariant measure. For $\mathrm{BMS}_3$, analogous structures arise with $\mathrm{Diff}^+(S^1)$ replacing $\mathrm{SL}(2,\mathbb{C})$.

1. Action: The group action is

$$U(M,T) \Phi[P] = e^{i \langle P, T \rangle} \rho(\tilde{B}(M, P)) \Phi[M^{-1} \cdot P]$$

where $\langle P, T \rangle$ is the natural dual pairing and $\tilde{B}(M, P)$ is the compensating little group element (Bekaert et al., 8 May 2025).

This construction applies mutatis mutandis to the extended BMS group (including superrotations), as well as for $\mathrm{BMS}_3$ and its higher/lower-dimensional analogues (Ruzziconi et al., 2 Jan 2026, Melas, 2017, Melas, 2021).

4. Representation Labels, Casimir Operators, and Orbit Structure

BMS UIRs are labeled by:

• The supermomentum orbit (including both “hard” Poincaré-like momentum and an infinite tower of genuine “supermomentum” modes).
• The choice of irreducible representation of the little group (generalizing spin/helicity to include infinite-dimensional or discrete data as appropriate).

The relevant Casimir operators are:

• Usual mass–shell quadratic Casimir $C_2 = \eta^{\mu\nu} p_\mu p_\nu$.
• For each $\ell \ge 2$, supertranslation Casimirs $C^{(\ell)} = \sum_{m=-\ell}^\ell |P_{\ell m}|^2$, and more generally, invariant polynomials in supermomentum (Chatterjee et al., 2017).
• For extended BMS, the set of superrotation charges and infinitely many constraints associated to the infinite symmetry algebra (Ruzziconi et al., 2 Jan 2026).

The dimensions of orbits and little groups control the spectral types and the decomposition structure:

• Orbits with full little groups (e.g., $\mathrm{SU}(2)$, $E(2)$) correspond to “hard” massive/massless multiplets.
• Generic orbits may have trivial or discrete little groups, leading to higher-dimensional orbits and BMS “particles” not of the conventional Poincaré type (Bekaert et al., 2024).

5. Wavefunctions, Quantum Vacua, and Hard/Soft Decomposition

Each BMS particle state can be realized as a quantum superposition of ordinary Poincaré plane waves, each propagating on a different “gravity vacuum”—specified by the soft supermomentum data. The general state is written as

$$|\Psi\rangle = \int_V \mathcal{D}C \int_{M_C} d^4X\; \Psi(X; \partial_z^2 C)\; |X; \partial_z^2 C\rangle$$

or, after Fourier transform,

$$|\Psi\rangle = \int \mathcal{D}N \int \omega d\omega\, d^2z\; \Psi(\omega, z; \partial_z^2 N) | \omega, z; \partial_z^2 N \rangle$$

where $|X; \partial_z^2 C\rangle$ indicates a basis labeled by “hard” position and “soft” vacuum (Bekaert et al., 2024, Bekaert et al., 8 May 2025).

There is a unique, Lorentz-invariant decomposition of a generic supermomentum into a sum of a hard (Poincaré) part and a soft part,

$$P(z, \bar{z}) = P_{\text{hard}}(z, \bar{z}) + \partial_z^2 \partial_{\bar{z}}^2 N(z, \bar{z})$$

with $N(z, \bar{z})$ encoding the net soft charge up to translations (Bekaert et al., 8 May 2025).

The same BMS state can yield different superpositions of Poincaré multiplets when projected onto distinct gravity vacua, reflecting the relativity of the concept of “particle” in the presence of an infinite degeneracy of the gravitational vacuum.

6. Branching, Tensor Products, and Gravitational Memory

Under restriction to the Poincaré subgroup, every BMS UIR of a given mass–shell branches into Poincaré UIRs with the same mass, and Poincaré spin or helicity labels determined by the embedding of the little group (Bekaert et al., 8 May 2025, Chatterjee et al., 2017). The decomposition is controlled by the structure of the BMS little group relative to the Poincaré little group.

Tensor products of BMS UIRs yield “dressed” outgoing states carrying the sum of the incoming supermomenta,

$$\varphi_{\text{total}} = \varphi_{(1)} + \varphi_{(2)}$$

Supermomentum conservation in gravitational scattering, i.e.,

$$\sum_{\text{in}} P_{\ell m}^{(\text{in})} = \sum_{\text{out}} P_{\ell m}^{(\text{out})}$$

enforced at the level of the S-matrix, gives a group-theoretic explanation for the gravitational memory effect: the persistent shift in the detector configuration at infinity due to passage of soft gravitational radiation (Chatterjee et al., 2017, Bekaert et al., 8 May 2025).

7. Extensions and Physical Significance

The extended BMS group, with superrotations, significantly enlarges the spectrum and structure of irreducible representations. For both three and four dimensions, extended BMS UIRs can be naturally realized as string-like (rather than point-like) objects: under Fourier transform, the wavefunction becomes a functional of an infinite set of coordinates associated with each supermomentum mode. The action of the group then mirrors that of worldsheet reparametrizations, and the carrier space is suggestively analogous to a space of fields on the celestial sphere or circle (Ruzziconi et al., 2 Jan 2026).

Physically, BMS UIRs with vanishing hard (Poincaré) momentum but nonzero supermomentum correspond to soft graviton configurations, degenerate gravitational vacua, or “soft hair” of black holes. In quantum gravity and S-matrix theory, their inclusion is required for completeness and to correctly capture the infrared sector and gravitational memory effects (Chatterjee et al., 2017, Bekaert et al., 8 May 2025).

Table: Key Features of BMS UIRs in 4D

Orbit Type	Little Group	Physical Interpretation
Massive (p² > 0)	SU(2)	Dressed massive particles
Massless, hard (p² = 0)	E(2)	Dressed massless particles, possible cont. spin
Soft (p = 0), SL(2,ℝ) stabilizer	SL(2,ℝ)	“dS₃-type” soft modes
Soft (p = 0), U(1) stabilizer	U(1)	Azimuthal soft modes, “soft hair”

• (Chatterjee et al., 2017) BMS symmetry, soft particles and memory
• (Bekaert et al., 2024) BMS particles
• (Bekaert et al., 8 May 2025) BMS representations for generic supermomentum
• (Amith et al., 20 Feb 2025) The massive BMS character in 3D quantum gravity
• (Melas, 2021) Representations of the Bondi-Metzner-Sachs group in three space-time dimensions in the Hilbert topology I. Determination of the representations
• (Barnich et al., 2015) Notes on the BMS group in three dimensions: II. Coadjoint representation
• (Ruzziconi et al., 2 Jan 2026) Extended BMS representations and strings
• (Melas, 2017) On the representation theory of the Bondi-Metzner-Sachs group and its variants in three space-time dimensions
• (Melas, 2013) Representations of the ultrahyperbolic BMS group HB.II. Determination of the representations induced from infinite little groups