Conformally Soft Photons and Gravitons (1810.05219v2)

Abstract: The four-dimensional $S$-matrix is reconsidered as a correlator on the celestial sphere at null infinity. Asymptotic particle states can be characterized by the point at which they enter or exit the celestial sphere as well as their $SL(2,\mathbb C)$ Lorentz quantum numbers: namely their conformal scaling dimension and spin $h\pm \bar h$ instead of the energy and momentum. This characterization precludes the notion of a soft particle whose energy is taken to zero. We propose it should be replaced by the notion of a conformally soft particle with $h=0$ or $\bar h=0$. For photons we explicitly construct conformally soft $SL(2,\mathbb C)$ currents with dimensions $(1,0)$ and identify them with the generator of a $U(1)$ Kac-Moody symmetry on the celestial sphere. For gravity the generator of celestial conformal symmetry is constructed from a $(2,0)$ $SL(2,\mathbb C)$ primary wavefunction. Interestingly, BMS supertranslations are generated by a spin-one weight $(\frac{3}{2},\frac{1}{2})$ operator, which nevertheless shares holomorphic characteristics of a conformally soft operator. This is because the right hand side of its OPE with a weight $(h,\bar h)$ operator ${\cal O}{h,\bar h}$ involves the shifted operator ${\cal O}{h+\frac{1}{2},\bar h+ \frac{1}{2}}$. This OPE relation looks quite unusual from the celestial CFT$_2$ perspective but is equivalent to the leading soft graviton theorem and may usefully constrain celestial correlators in quantum gravity.

Insightful Overview of "Conformally Soft Photons and Gravitons"

The paper "Conformally Soft Photons and Gravitons" presents an innovative approach to redefining aspects of quantum field theory in the context of celestial conformal field theory (CFT). This work explores how four-dimensional scattering amplitudes can be expressed as correlators on the celestial sphere at null infinity, introducing a new characterization of soft particles as "conformally soft" particles. The essence of this work lies in the reformulation of asymptotic particle states, traditionally characterized by energy and momentum, in terms of their conformal scaling dimension and spin within the celestial CFT.

Celestial Sphere Scattering Framework

The authors start by challenging the conventional notion of soft particles, whose energy approaches zero. Instead, they introduce "conformally soft" particles associated with zero conformal dimensions ($h=0$ or $\bar{h}=0$). This new taxonomy is foundational for analyzing interactions on the celestial sphere, leveraging $SL(2,\mathbb C)$ Lorentz quantum numbers. The proposed framework replaces the energy-momentum eigenstate representation with conformal primary wavefunctions labeled by their dimensions and positions on the celestial sphere.

Construction of Conformally Soft Particles

Key contributions include constructing conformally soft modes for both photons and gravitons. For photons, the authors derive conformally soft currents with dimensions (1,0) and identify them as generators of a $U(1)$ Kac-Moody symmetry on the celestial sphere. In the gravitational context, conformally soft graviton modes are linked to celestial conformal symmetry generated by a (2,0) primary wavefunction. These developments demonstrate the adaptability of asymptotic symmetries like BMS supertranslations in this conformal framework.

Analytical Insights and OPE Relations

One significant analytical insight is the unusual OPE relation introduced by the paper, where the operator product expansion (OPE) involving the spin-one weight $(\frac{3}{2}, \frac{1}{2})$ operator shifts the target operator dimensions by `$$(\frac{1}{2}, \frac{1}{2})`. This seems unconventional from a celestial CFT$$_2$ perspective but aligns with the leading soft graviton theorem, providing a potentially useful constraint on celestial correlators pertinent to quantum gravity.

Theoretical and Practical Implications

The implications of this research are twofold: theoretically, it foregrounds celestial CFT as a robust framework for understanding asymptotic symmetries and soft theorems in quantum gravity. Practically, this characterization can influence how scattering amplitudes are analyzed, offering novel insights into symmetries and interactions. Future developments could further enhance the role of celestial conformal field theories in addressing fundamental questions in high-energy physics and quantum gravity, potentially leading to new methods for analyzing celestial correlators.

Speculations on AI and Future Directions

In light of the remarkable mathematics involved, computational advancements, particularly in AI, could facilitate the exploration of these conformal structures. Machine learning algorithms could be developed to simulate or predict outcomes of scattering processes on the celestial sphere, offering new insights into quantum gravity models.

The paper "Conformally Soft Photons and Gravitons" is a noteworthy commitment towards integrating celestial conformal field theory and asymptotic symmetries, marking a significant step in rethinking quantum field theory frameworks and contributing fresh methodologies in theoretical physics with robust mathematical underpinnings.