BMS supertranslations and Weinberg's soft graviton theorem (1401.7026v2)

Abstract: Recently it was conjectured that a certain infinite-dimensional "diagonal" subgroup of BMS supertranslations acting on past and future null infinity (${\mathscr I}^-$ and ${\mathscr I}^+$) is an exact symmetry of the quantum gravity ${\cal S}$-matrix, and an associated Ward identity was derived. In this paper we show that this supertranslation Ward identity is precisely equivalent to Weinberg's soft graviton theorem. Along the way we construct the canonical generators of supertranslations at ${\mathscr I}^\pm$, including the relevant soft graviton contributions. Boundary conditions at the past and future of ${\mathscr I}^\pm$ and a correspondingly modified Dirac bracket are required. The soft gravitons enter as boundary modes and are manifestly the Goldstone bosons of spontaneously broken supertranslation invariance.

• The paper establishes that diagonal BMS supertranslations yield a Ward identity equivalent to Weinberg's soft graviton theorem, linking symmetry to scattering amplitudes.
• The authors employ boundary conditions and modified Dirac brackets to define a phase space that includes soft graviton modes as effective Goldstone bosons.
• The work unifies classical gravitational scattering techniques with quantum frameworks, paving the way for future research in quantum gravity.

Overview of "BMS Supertranslations and Weinberg's Soft Graviton Theorem"

The paper by Temple He, Vyacheslav Lysov, Prahar Mitra, and Andrew Strominger establishes a profound connection between general relativity's symmetries and quantum gravity through analysis of the BMS (Bondi–van der Burg–Metzner–Sachs) supertranslations and Weinberg's soft graviton theorem. The authors conjecture and demonstrate that a particular infinite-dimensional subgroup, referred to as diagonal BMS supertranslations, is an exact symmetry of the quantum gravity S-matrix, showing its equivalence with the soft graviton theorem.

Key Contributions

The paper's core contribution lies in the assertion and evidence that the identified BMS supertranslation symmetry leads to a Ward identity that aligns precisely with Weinberg's soft graviton theorem. This theorem universally links any S-matrix element involving a graviton of diminishing four-momentum to a second matrix element without such a graviton. This symmetry's universality is especially telling, given that it does not depend on the spin or other quantum characteristics of the asymptotic particles involved.

Theoretical Insights

1. BMS Supertranslations as Symmetry: The paper constructs canonical generators for these supertranslations at past and future null infinity, denoted as $\mathcal{I}^\pm$, showing that their Ward identities are equivalent to Weinberg’s soft graviton theorem. This finding connects long-standing results about Minkowski scattering structure in gravitational theories to these symmetries.
2. Boundary Conditions and Dirac Brackets: The convergence to an exact interpretation required defining physical phase spaces at $\mathcal{I}^\pm$, incorporating boundary conditions, and modifying the Dirac bracket. Here, soft gravitons are interpreted as boundary modes, effectively Goldstone bosons, hinting at the spontaneous breaking of supertranslation invariance.
3. Conformal Compactification and Phase Space: The authors illustrate that the total phase space must account for both usual radiative modes and additional soft modes. To manage this, they implement constraints at the boundaries of null infinities, solutions that factor crucially into how modes interact in the S-matrix.

Numerical Results and Claims

The paper stops short of providing numerical calculations or results common in empirical studies because it focuses extensively on theoretical proofs and symmetry explorations. Importantly, it suggests that the matching between the soft graviton theorem and the -matrix Ward identities could apply equally in non-vacuum and gauge field theories, broadening the scope of this theoretical framework.

Implications and Future Directions

This work expands the understanding of how classical gravitational symmetries manifest within quantum frameworks, specifically when linked to scattering processes. By broadening the BMS framework to include quantum scattering, the authors place traditional gravitational scattering techniques within a quantum gravity context.

The convergence of quantum and classical theories under such symmetries may guide future research in developing quantum gravity theories, potentially improving insights into gravitational wave observables. Moreover, the findings might stimulate further exploration on the potential inclusion of other massive fields and gauge theories within this framework.

Conclusion

He, Lysov, Mitra, and Strominger’s paper achieves significant theoretical progress by uniting a well-regarded classical theorem about gravitational interactions (the soft graviton theorem) with a broader symmetry framework derived from BMS considerations. It elevates the understanding of asymptotic symmetries at null infinity, showing practical implications for both theoretical foundations and potential future experimental inquiries within quantum gravity. The paper thus represents an important conceptual advance in the pursuit of a cohesive, symmetry-driven understanding of gravitational scattering and quantum gravity dynamics.