Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere (1701.00049v1)

Abstract: The four-dimensional (4D) Lorentz group $SL(2,\mathbb{C})$ acts as the two-dimensional (2D) global conformal group on the celestial sphere at infinity where asymptotic 4D scattering states are specified. Consequent similarities of 4D flat space amplitudes and 2D correlators on the conformal sphere are obscured by the fact that the former are usually expressed in terms of asymptotic wavefunctions which transform simply under spacetime translations rather than the Lorentz $SL(2,\mathbb{C})$. In this paper we construct on-shell massive scalar wavefunctions in 4D Minkowski space that transform as $SL(2,\mathbb{C})$ conformal primaries. Scattering amplitudes of these wavefunctions are $SL(2,\mathbb{C})$ covariant by construction. For certain mass relations, we show explicitly that their three-point amplitude reduces to the known unique form of a 2D CFT primary three-point function and compute the coefficient. The computation proceeds naturally via Witten-like diagrams on a hyperbolic slicing of Minkowski space and has a holographic flavor.

• The paper constructs on-shell massive scalar wavefunctions that transform as SL(2,C) conformal primaries, recasting traditional scattering amplitudes.
• It establishes a direct connection between 4D amplitudes and 2D conformal field theory, demonstrating the role of Witten-like diagrams and mass relations.
• The study explicitly computes massive three-point functions under specific mass conditions, highlighting potential implications for holographic quantum gravity.

On Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere

The paper "Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere" authored by Sabrina Pasterski, Shu-Heng Shao, and Andrew Strominger explores the connection between four-dimensional (4D) Lorentz symmetry and two-dimensional (2D) conformal field theories (CFT) via a novel approach to viewing scattering amplitudes in flat space. This exploration leverages the celestial sphere at null infinity as a platform for re-imagining the description of 4D quantum field theory (QFT) processes.

Summary of Contributions

This work focuses on reframing 4D Minkowski space scattering amplitudes in terms of conformal structures. Traditional representations often obscure the $SL(2,\mathbb{C})$ invariance due to the basis of asymptotic wavefunctions, which emphasize spacetime translational symmetry. This paper proposes an alternative by constructing massive scalar wavefunctions that are on-shell in this space and behave as $SL(2,\mathbb{C})$ conformal primaries.

Key Insights

1. Reparameterization of Wavefunctions: The construction involves deriving on-shell massive scalar wavefunctions in 4D space, allowing them to transform under $SL(2,\mathbb{C})$ rather than the usual asymptotic plane wave solutions. This presents the scattering amplitudes as conformal covariant, offering a new mathematical framework for quantum field theories that can seamlessly interface with gravity.
2. Relation to 2D CFT: The treatment translates certain mass relations into well-known three-point conformal field functions, highlighting similarities between flat space scattering and 2D CFT behavior. This connection is strengthened through Witten-like diagrams on hyperbolic slicing of Minkowski space, proposing a holographic flavor to the adaptation of these amplitudes.
3. Massive Three-Point Functions: For specific mass conditions, the three-point amplitude takes on a closed form congruent with known forms in 2D CFT, with the paper providing an explicit computation for the coefficient in such cases. This suggests deeper underlying symmetries and potentials for extending conformal methods to broader physics applications.

Implications and Future Outlook

The theoretical implications of this work extend towards understanding gravity and gauge theory scattering processes in terms of boundary conformal theories. By aligning 4D scattering amplitudes in terms of 2D CFT constructs, it opens pathways to interpreting interactions in potentially holographic terms, adhering to various duality conjectures that suggest 4D quantum gravitational theories might possess 2D boundary descriptions. While the paper refrains from claiming progress in quantum gravity directly, the tools and framework set forth seem poised for more extensive exploitation in that domain.

Future research may explore the computational adaptability and further numerical verifications of the proposed covariant wavefunctions against traditional models. Additionally, investigations might delve into specific gauge theories or superconformal models for elucidation of symmetry roles in amplitude simplification, especially in already mathematically rich frameworks like $\mathcal{N}=4$ super-Yang-Mills.

In conclusion, Pasterski, Shao, and Strominger's paper provides a robust mathematical paradigm for the propagation and transformation of quantum states at infinity, using the celestial sphere model. The insights on conformal symmetries offer new perspectives that can potentially bridge conceptual divides between seemingly disparate areas of high-energy theoretical physics.