Lectures on Celestial Amplitudes (2108.04801v1)

Abstract: Lecture notes prepared for the 2021 SAGEX PhD School in Amplitudes hosted by the University of Copenhagen August 10th through 13th. Topics covered include: the manifestation of asymptotic symmetries via soft theorems, their organization into currents in a celestial CFT, aspects of the holographic dictionary, a literature guide, and accompanying exercises.

• The paper demonstrates how celestial amplitudes reframe the gravitational S-matrix as conformal correlators on the celestial sphere.
• It employs integral transforms to connect momentum-space scattering with celestial CFT structures and soft theorem identities.
• The study shows that embedding UV data in celestial amplitudes offers fresh insights into quantum gravity and the resolution of infrared divergences.

Overview of "Lectures on Celestial Amplitudes"

The document titled "Lectures on Celestial Amplitudes," authored by Sabrina Pasterski, serves as a comprehensive set of lecture notes, targeting fundamental aspects of celestial amplitudes and their role within the framework of celestial holography. Presented during the 2021 SAGEX PhD School in Amplitudes, these notes focus on connecting the manifestations of asymptotic symmetries in gravitational scattering to celestial Conformal Field Theory (CFT) structures.

Key Concepts and Structure

Celestial holography draws inspiration from the AdS/CFT correspondence, proposing a duality between scattering amplitudes in asymptotically flat spacetimes and a CFT situated at the celestial sphere. Central to this framework are celestial amplitudes, which reconceptualize the $\mathcal{S}$-matrix elements within a basis that adheres to conformal correlator properties on the celestial sphere.

The lecture notes encapsulate several pivotal themes:

1. Asymptotic Symmetries and Soft Theorems: These address how infrared physics engenders infinite-dimensional symmetry enhancements, offering a glimpse into a holographic description via soft theorems.
2. Holographic Dictionary: The mapping is established between $\mathcal{S}$-matrix elements and conformal correlators, allowing the reinterpretation of known amplitudes through conformal transformations using a set of integral transforms.
3. Scattering Set-up: This includes a detailed examination from momentum space to spacetime perspectives, reviewing global symmetries, asymptotic symmetries, and the intrinsic connections to soft theorems and memory effects.
4. Conformal Primary States: The construction of conformal primary wavefunctions, identifying redundancy via shadow transforms, and addressing the conditions under which these modes become pure gauge.
5. Ward Identities and Soft Theorems: Establish the equivalence between soft theorems and asymptotic symmetry Ward identities, employing the underlying celestial CFT framework.
6. OPE Structure: Celestial OPEs capture collinear limits, with the symmetries constraining their leading coefficients, enabled by the construction of celestial diamonds to explore recursion relations.

Numerical Results and Implications

Celestial amplitudes probe all energy scales in scattering, thereby inverting the Wilsonian paradigm. This paradigm shift implies that rather than breaking high-energy physics into effective theories, the amplitudes encode UV data intrinsically. For instance, 4-graviton scattering amplitudes demonstrate divergences remedied by string form factors, emphasizing celestial amplitudes' sensitivity to UV properties.

Theoretical Implications and Future Directions

The reformulation of $\mathcal{S}$-matrix dynamics within this celestial holographic posture could elucidate new features of quantum gravity, particularly enhancing the understanding of infrared divergences. Furthermore, it posits an innovative framework for investigating black hole information paradoxes through asymptotic symmetries, also hinting at broader horizons for quantum gravity quantization.

The lecture notes underscore the imminent trajectory for further research—extending these discussions to encompass massive fields, supersymmetric theories, and higher spin counterparts. They accentuate the requirement for an explicit celestial state-operator correspondence, which could culminate in an unambiguous celestial CFT formulation, potentially extending to all perturbative quantum field theories.

In conclusion, these notes stitch a vivid mosaic merging celestial representations and scattering amplitudes, sketching a path that's deeply interwoven with the fabric of quantum field theories, laying fertile grounds for theoretical advancements in the landscape of holography and quantum gravity.