Infrared physics of QED and gravity from representation theory

Abstract: The infrared structure of QED and gravity is known to be governed by an infinite-dimensional symmetry group which extends the Poincaré group to include, respectively, large $U(1)$ transformations and BMS supertranslations. We describe how the unitary irreducible representations (UIRs) of these asymptotic symmetry groups encode universal infrared features of a scattering process. Motivated by the goal of defining an infrared-finite $S$-matrix based on these UIRs, we also study supermomentum eigenstates and contrast our construction with the dressed-state approach for infrared-safe amplitudes.

• The paper demonstrates that organizing asymptotic states as UIRs of extended symmetry groups resolves infrared divergences in QED and gravity.
• It introduces a hard/soft decomposition of supermomenta and establishes unique norms for soft sectors, ensuring IR-finite amplitudes.
• The work presents a unified framework linking soft theorems, memory effects, and the construction of dressed states for unitary scattering matrices.

Representation Theory Approach to Infrared Physics in QED and Gravity

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Motivation and Background

Quantum Electrodynamics (QED) and perturbative gravity are plagued by infrared (IR) divergences in scattering amplitudes involving massless bosons. Traditional treatments, such as resummation of divergent terms and inclusive cross-section calculations, control such divergences at the level of physically measurable observables but do not yield a mathematically well-defined, unitary $S$-matrix on the conventional Fock space. Previous works, notably the Faddeev-Kulish (FK) framework and its generalizations, construct dressed states—charged particles accompanied by coherent clouds of soft bosons—which lead to IR-finite scattering amplitudes; however, such states exhibit ambiguities, lack a clear status as normalizable Fock states, and fail to systematically encode the conserved quantum numbers associated with asymptotic symmetries.

Over the last decade, profound insights have emerged connecting soft theorems, memory effects, and symmetries at null infinity to infinite-dimensional extensions of the Poincaré group: large $U(1)$ gauge transformations in QED and the Bondi-Metzner-Sachs (BMS) group in gravity. These symmetry groups impose infinitely many conservation laws in the infrared regime, suggesting that asymptotic states should be organized according to unitary irreducible representations (UIRs) of these extended symmetry algebras.

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Structure of Asymptotic Symmetry Groups

QED Asymptotic Symmetry:

The group is $$SO(3,1) \ltimes (\mathbb{R}^{3,1} \times \mathcal{E}[0])$$, where $\mathcal{E}[0]$ denotes smooth weight-zero conformal densities on the celestial sphere. It extends translations to include angle-dependent large $U(1)$ transformations.

Gravity Asymptotic Symmetry (BMS):

The group is $SO(3,1) \ltimes \mathcal{E}[1]$, where $\mathcal{E}[1]$ are smooth weight-one conformal densities (supertranslations). The Lorentz group acts naturally, and supertranslations generalize spacetime translations.

The representation theory for groups of the form $SO(3,1) \ltimes A$ (with abelian $A$) uses induced representations parameterized by elements of the dual space $A^*$ (supermomenta), leading to a classification in terms of Lorentz orbits and associated little groups.

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Hard and Generic Representations

Hard Representations

Hard UIRs correspond to conventional Poincaré particles lifted to the larger asymptotic symmetry group. The hard supermomentum for each particle is completely determined by physical momentum and charge, with explicit forms:

• QED (for a scalar of momentum $p^\mu$ and charge $q_e$):
  • Massless: $$\Big(p^\mu, Q(z,)\Big) = (\omega q^\mu(\zeta, \bar{\zeta}), q_e \delta^{(2)}(z-\zeta))$$
  • Massive: $$\Big(p^\mu, Q(z,)\Big) = (p^\mu, \dfrac{q_e m^2}{4\pi\, (q(z,)\cdot p)^2})$$
• BMS (gravity, scalar momentum $p^\mu$):
  • Massless: $P(z,) = \omega \delta^{(2)}(z-\zeta)$
  • Massive: $P(z,) = -\dfrac{m^4}{4\pi (q(z,)\cdot p)^3}$

Hard UIRs have maximal-dimensional little groups coinciding with the standard Poincaré little groups.

Generic Representations

Generic UIRs allow for non-trivial soft supermomentum components and reduced little groups. The supermomentum can be uniquely decomposed into "hard" and "soft" pieces:

• QED: $$P = (p_\mu, Q(z,)) + (0_\mu, \eth\bar{\eth}\mathcal{N}(z,))$$
• BMS: $$P(z,) = P_{\text{hard}}(z,) + \eth^2\bar{\eth}^2 N(z,)$$

The decomposition is nonlinear and Lorentz-invariant. The soft sector forms a Hilbert space with explicit invariant norms:

• QED: $$\|\mathcal{N}\|^2 = \int d^2z\,\partial_z\mathcal{N}\partial_{\bar{z}}\mathcal{N}$$
• BMS: $$\|N\|^2 = \int d^2z\,\partial^2_z N\,\partial^2_{\bar{z}} N$$

These spaces admit a two-point function encoding the inner product structure via projective null cone contractions.

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Infrared Divergences as Supermomentum Conservation Failure

In conventional scattering processes involving only hard states, conservation of momentum and charge does not guarantee conservation of supermomentum due to nonlinear distributional identities. The obstruction is directly given by the soft factors in soft photon/graviton theorems:

• QED Soft Factor: $$\mathcal{S}(z,) = \frac{1}{2\pi}\sum_i \eta_i q_i \ln|p_i \cdot q(z,)|$$
• Gravity Soft Factor: $$\mathscr{S}(z,) = -\frac{1}{2\pi}\sum_i \eta_i (p_i \cdot q(z,)) \ln|p_i \cdot q(z,)|$$

These soft factors are precisely the missing terms required for supermomentum conservation. The formalism recasts soft theorems as Ward identities of asymptotic symmetry groups.

Exponentiation Formula for Virtual Divergences:

The real part of the exponent in the IR factorization theorem is proportional to the norm squared of the soft factor:

• QED: $$\Re(\mathcal{W}) = -\frac{\log(\Lambda/\lambda)}{8\pi}\|\mathcal{S}\|^2$$
• Gravity: $$\Re(\mathcal{W}) = -2G\log(\Lambda/\lambda)\|\mathscr{S}\|^2$$

For gravity, the formula is well-behaved even in the presence of massless particles, while QED suffers additional collinear divergences for massless charged particles.

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Dressed States and Supermomentum Eigenstates

Faddeev-Kulish (FK) and Generalized Dressings

Dressing operators attach coherent clouds of soft bosons to charged particles, ensuring IR-finiteness. The dressing ambiguity—due to gauge freedom—manifests as freedom in the dressing profile (vector $c^\mu$ for QED, tensor $c^{\mu\nu}$ for gravity).

• QED FK Dressing:

$|\vec{p}\rangle_{FK} = e^{\hat{R}}|\vec{p}\rangle$, where $\hat{R}$ is constructed from a profile $$f^\mu(k,p) = q_e\left(\frac{p^\mu}{p\cdot k} - c^\mu\right)\psi(k,p)$$.

• Gravity FK Dressing:

$|\vec{p}\rangle_{FK} = e^{\hat{R}}|\vec{p}\rangle$, with $$f^{\mu\nu}(k,p) = \left(\frac{p^\mu p^\nu}{p\cdot k} + c^{\mu\nu}\right)\psi(k,p)$$.

Dressed states are eigenstates of the soft charge, but not of the full supermomentum operator unless a singular limiting procedure is implemented. The dressing can be adjusted to yield supermomentum eigenstates, thereby linearizing the supermomentum in the momenta and ensuring conservation for suitable choices.

Supermomentum Eigenstates and Goldstone Operators

Through an appropriate limit (regulating the dressing profile), dressed states can be mapped to true supermomentum eigenstates. This procedure is mathematically akin to introducing a canonical Goldstone operator and promoting the soft sector to a representation-theoretic basis. The Goldstone operator realizes the canonical commutation relation with the soft charge ($[\hat{Q},\hat{\Phi}] = -i$ in QED; similar for gravity).

The resulting supermomentum eigenstates have the property that, if all dressings are taken appropriately, the nonlinear obstruction vanishes and conservation of momentum and charge directly implies supermomentum conservation. IR-finite amplitudes can thus be systematically constructed.

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Implications and Outlook

The representation-theoretic approach provides a unified and rigorous language for encoding the universal infrared structure of scattering in QED and gravity. By organizing asymptotic states in terms of UIRs of the full asymptotic symmetry group, the formalism naturally incorporates the infinite set of conservation laws and provides a well-defined Hilbert space structure, circumventing ambiguities and pathologies of previous constructions.

Numerical Results and Claims:

• The real part of the virtual IR divergence exponent is exactly proportional to the invariant norm of the obstruction to supermomentum conservation.
• All IR-safe dressed states correspond to supermomentum eigenstates under suitable limiting procedures.
• The nonlinear hard/soft decomposition of supermomenta is unique and invariant, distinct from spherical harmonics decompositions.

Contradictory Claims:

• Not all dressed states are equivalent: different choices of dressing (gauge ambiguities) can alter matrix elements and analytic properties, as documented in recent works.
• Conventional Fock states are insufficient for IR-finite $S$-matrix elements; only the broader Hilbert space of induced representations guarantees unitary asymptotic dynamics.

Practical and Theoretical Implications:

• Offers a systematic platform for constructing IR-finite, unitary $S$-matrices in both QED and perturbative gravity.
• Facilitates a mathematically robust formulation of memory effects and their quantum counterparts.
• Provides a foundation for potential developments in flat-space holography, with correspondence between boundary symmetry representation theory and bulk amplitudes.
• Pathways to extend to higher dimensions, nonabelian gauge theories, and enlarged symmetry groups (e.g., superrotations).

Speculation on Future Directions:

• A new particle basis, defined via UIRs of asymptotic symmetry groups, may replace Poincaré-based definitions in IR-sensitive theories.
• Extension of the method to non-abelian gauge theories may require major conceptual advances in induced representation theory.

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Conclusion

The paper demonstrates that a systematic representation-theoretic classification of asymptotic symmetry groups ($SO(3,1) \ltimes$ infinite-dimensional abelian group) encodes the entire infrared physics of QED and gravity in flat spacetime. The framework unifies soft theorems, memory effects, and IR cancellation in scattering amplitudes, directly linking the non-conservation of supermomentum for conventional states to the universal structure of IR divergences. The notion of asymptotic particles as UIRs of the extended symmetry provides a mathematically well-defined basis for both practical calculations and theoretical understanding, with significant implications for the structure of quantum field theory beyond standard Fock space and the ongoing development of flat-space holography. Finite, unitary scattering requires an enlarged Hilbert space constructed from these induced representations, and the approach outlined lays the groundwork for systematic advances in IR physics across both gauge and gravitational theories (2603.06297).