A Conformal Basis for Flat Space Amplitudes (1705.01027v1)

Abstract: We study solutions of the Klein-Gordon, Maxwell, and linearized Einstein equations in $\mathbb{R}^{1,d+1}$ that transform as $d$-dimensional conformal primaries under the Lorentz group $SO(1,d+1)$. Such solutions, called conformal primary wavefunctions, are labeled by a conformal dimension $\Delta$ and a point in $\mathbb{R}^d$, rather than an on-shell $(d+2)$-dimensional momentum. We show that the continuum of scalar conformal primary wavefunctions on the principal continuous series $\Delta\in \frac d2+ i\mathbb{R}$ of $SO(1,d+1)$ spans a complete set of normalizable solutions to the wave equation. In the massless case, with or without spin, the transition from momentum space to conformal primary wavefunctions is implemented by a Mellin transform. As a consequence of this construction, scattering amplitudes in this basis transform covariantly under $SO(1,d+1)$ as $d$-dimensional conformal correlators.

• The paper establishes that conformal primary wavefunctions form a complete basis for solving flat space wave equations for both massless and massive fields.
• It employs Mellin transforms and Lorentz group techniques to map scattering amplitudes onto a conformal field theory framework.
• The study extends its analysis to higher spin fields, offering key insights with implications for holography and quantum gravity computations.

An Analysis of "A Conformal Basis for Flat Space Amplitudes"

The paper "A Conformal Basis for Flat Space Amplitudes" by Sabrina Pasterski and Shu-Heng Shao explores the intricate relationship between scattering amplitudes and the conformal primary wavefunctions, which are significant in conformal field theory (CFT) and quantum field theory (QFT). The authors establish a framework for describing solutions to the Klein-Gordon, Maxwell, and linearized Einstein equations using conformal primary wavefunctions that are characterized by a conformal dimension and a point in $\mathbb{R}^d$.

Key Contributions

The central contribution of the paper is constructing a conformal basis for scattering amplitudes in flat spacetime, which elegantly connects the physics of scattering processes to the mathematical structures of conformal field theories. The major highlights are:

1. Conformal Primary Wavefunctions: The authors rigorously define solutions to wave equations as conformal primary wavefunctions, transforming them as conformal primaries under Lorentz group $SO(1,d+1)$, which is crucial for understanding their role in scattering processes.
2. Completeness in the Principal Continuous Series: They demonstrate that scalar conformal primary wavefunctions on the principal continuous series form a complete basis for the solutions to the wave equations. Notably, the massless case utilizes the Mellin transform to transition from momentum space to the conformal primary wavefunctions.
3. Extension to Spin: The manuscript extends the construction of conformal primary wavefunctions to fields with spin, incorporating spin-one (photon) and spin-two (graviton) fields into the framework. This work is critical for advancing theoretical predictions in gauge theories and general relativity.
4. Implications for Holography: By leveraging the isomorphism between the Lorentz group and conformal group, this research sheds light on potential holographic descriptions of scatterings, echoing the holographic principle observed in gravity theories.

Numerical and Formal Results

With substantial technical rigor, the authors establish that for both massive and massless fields, conformal primary wavefunctions within the specific range of conformal dimensions provide a robust basis for quantum field theory calculations. They verify completeness through orthogonality and normalizability conditions, confirming the foundational nature of the principal continuous series in these analyses.

Implications and Future Directions

This work has profound implications for both theoretical and practical aspects. Theoretically, it enhances the conceptual bridge between flat-space physics and conformal field theories; practically, it proposes a new computation framework in scattering amplitude studies, integral to particle physics research. The insights gained here can potentially lead to refined methods for computing amplitudes in quantum gravity and string theory. Future directions might explore further applications in non-trivial spacetimes and more complex interaction models.

For the community of advanced researchers, this paper presents a nuanced approach to understanding scattering amplitudes through the lens of conformal symmetry, heralding new methodologies in the exploration of the quantum field.