Polyharmonic green functions and nonlocal Bondi-Metzner-Sachs transformations of a free scalar field

We express the nonlocal Bondi-Metzner-Sachs (BMS) charges of a free massless Klein-Gordon scalar field in 2 þ 1 in terms of the Green functions of the polyharmonic operators. Using the properties of these Green functions, we are able to discuss the asymptotic behavior of the fields that ensures the existence of the charges and prove that one obtains a realization of the 2 þ 1 BMS algebra in canonical phase space. We also discuss the transformations in configuration space and show that in this case the algebra closes only up to skew-symmetric combinations of the equations of motion. The formulation of the charges in terms of Green functions opens the way to the generalization of the formalism to other dimensions and systems.
We acknowledge discussions with Marc Henneaux and Axel Kleinschmidt. The work of C. B. is partially supported by the Project MASHED (Grant No. TED2021-129927BI00), funded by Grant No. MCIN/AEI/10.13039/ 501100011033 and by the European Union Next Generation EU/PRTR. J. G. has been supported in part by Grants No. MINECO FPA2016-76005-C2-1-P and No. PID2019-105614 GB-C21 and from the State Agency for Research of the Spanish Ministry of Science and Innovation through the Unit of Excellence Maria de Maeztu 2020–2023 award to the Institute of Cosmos Sciences (Grant No. CEX2019-000918-M).
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Postprint (published version)