Conformal properties of soft operators: Use of null states


Soft operators are (roughly speaking) zero energy massless particles which live on the celestial sphere in Minkowski space. The Lorentz group acts on the celestial sphere by conformal transformation and the soft operators transform as conformal primary operators of various dimension and spin. Working in space-time dimensions D=4 and 6, in this paper, we study some properties of the conformal representations with (leading) soft photon and graviton as the highest weight operators. Typically these representations contain null vectors. We argue, from the S-matrix point of view, that infinite dimensional asymptotic symmetries and conformal invariance require us to set these null vectors to zero. As a result, the corresponding soft operator satisfies linear partial differential equation (PDE) on the celestial sphere. Curiously, these PDEs are equations of motion of Euclidean gauge theories on the celestial sphere with scalar gauge invariance, i.e., the gauge parameter is a scalar field on the sphere. These are probably related to large U(1) and supertranslation transformations at infinity. Now, the PDE satisfied by the soft operator can be converted into PDE for the S-matrix elements with the insertion of the soft operator. These equations can then be solved subject to appropriate boundary conditions on the celestial sphere, provided by (Lorentz) conformal invariance. The resulting soft S-matrix elements have an interesting “pure-gauge” form and are determined in terms of a single scalar function. Heuristically speaking, the role of the null state decoupling is to reduce the number of degrees of freedom or polarization states of soft photon and graviton to one, given effectively by a single scalar function. This reduction in the number of degrees of freedom makes the Ward identity for the asymptotic symmetry almost integrable. The result of the integration, which we are not able to perform completely, should of course be Weinberg’s soft theorem. Finally, we comment on the resemblance of all of these things to quantization of fundamental strings. Read More

Publication: American Physical Society – Physical Review D

Publisher: ResearchGate

Authors: Shamik Banerjee, Pranjal Pandey and Partha Paul

Keywords: Soft operators, dimensional asymptotic symmetries, conformal invariance, conformal transformation

Meet one of the Author:

Dr Pranjal Pandey earned his Ph.D. in Theoretical High Energy Physics from the Homi Bhabha National Institute, which is under the aegis of the Department of Atomic Energy in India. His publication on the topic of Flat Space Holography, has been cited by leading groups at Harvard, Princeton, and Oxford. His extensive experience in using mathematical models to price derivatives and forecast market movements based on alternative text data, like tweets and news headlines, distinguishes him from others.

Dr. Pranjal Pandey


Institute of Physics, Sachivalaya Marg, Bhubaneshwar, 751005, India – The Institute of Physics in Bhubaneswar, India is a research institution that is funded jointly by the Department of Atomic Energy (DAE) and the government of Odisha. It was officially established in 1972 and began its academic activities in 1975 with two faculties and the director. From 1974 to 1981, the institute operated out of rented buildings before moving to its current campus in 1981. In 1985, the institute was taken over by the DAE. The predoctoral program at the institute provides a broad-based education in advanced physics and research methodology. Upon completion of the program, students are eligible to pursue a Ph.D. degree awarded by the Homi Bhabha National Institute of the DAE.

Homi Bhabha National Institute, Anushakti Nagar, Mumbai, 400085, India – The Homi Bhabha National Institute (HBNI) was founded in 2005 under the University Grants Commission Act. It has several constituent institutions and off-campus centers. HBNI’s role is to cultivate expertise in nuclear science and engineering and to encourage basic research that can be translated into technological development and applications through academic programs, such as Master’s and Ph.D. degrees in various fields of engineering, physical, chemical, mathematical, life, and medical and health sciences. HBNI also promotes interdisciplinary research.

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