Abstract
The formalism of relativistic partial wave expansion is developed for four-point celestial amplitudes of massless external particles. In particular, relativistic partial waves are found as eigenfunctions to the product representation of celestial Poincaré Casimir operators with appropriate eigenvalues. The requirement of hermiticity of Casimir operators is used to fix the corresponding integral inner product, and orthogonality of the obtained relativistic partial waves is verified explicitly. The completeness relation, as well as the relativistic partial wave expansion follow. Example celestial amplitudes of scalars, gluons, gravitons and open superstring gluons are expanded on the basis of relativistic partial waves for demonstration. A connection with the formulation of relativistic partial waves in the bulk of Minkowski space is made in appendices.
Highlights
Theorems familiar from Minkowski space amplitudes correspond to so called conformal soft theorems for celestial amplitudes, which were studied in [22,23,24,25,26,27,28,29,30]
The formalism of relativistic partial wave expansion is developed for four-point celestial amplitudes of massless external particles
As we proposed in [35], instead of conformal partial wave decomposition we consider relativistic partial wave decomposition, the derivation of which on the celestial sphere is the subject of this work
Summary
Since celestial amplitudes are set up such that they transform as correlators of conformal primaries under Lorentz transformations, one is led to consider the 2D (global) conformal group representation theory on the celestial sphere, which in the case at hand corresponds to labeling exchanged auxiliary modes by the characteristic numbers conformal dimension: ∆ , and spin: J. Whether we choose to use one dual energy variable or a pair, from the point of view of Poincaré group representation theory the resulting conformal primary wavefunction parametrizes a mode of characteristic numbers m = 0 and helicity J. infinity of an asymptotically flat space-time instead of Minkowski space, the Poincaré group is superseded by the (extended) BMS group [6,7,8]. This makes the study of irreducible representations of 4D extended BMS group in Minkowski signature a very important task, being a prerequisite for understanding orthogonal modes in theories with extended BMS symmetry
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